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Accumulation functions practice problems

Express an accumulation function ∫ₐˣ f(t) dt as a function of x, and check it against FTC Part 1. 54 problems with worked solutions; in 54 of them every equation is proved by a computer algebra system.

14 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 4.1077

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} e^{t} \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} e^{t} = e^{t} \]
    An antiderivative of the integrand is exp(t).✓ Proved
  2. \[ - \left. e^{t} \right|_{\substack{ t=1 }} + \left. e^{t} \right|_{\substack{ t=x }} = e^{x} - e \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(e^{x} - e\right) = e^{x} \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. e^{x} - e \right|_{\substack{ x=1 }} = 0 \]
    And F(1) = 0.✓ Proved
Answer \( F(x) = e^{x} - e \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1079

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} e^{t} \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} e^{t} = e^{t} \]
    An antiderivative of the integrand is exp(t).✓ Proved
  2. \[ - \left. e^{t} \right|_{\substack{ t=-1 }} + \left. e^{t} \right|_{\substack{ t=x }} = e^{x} - e^{-1} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(e^{x} - e^{-1}\right) = e^{x} \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. e^{x} - e^{-1} \right|_{\substack{ x=-1 }} = 0 \]
    And F(-1) = 0.✓ Proved
Answer \( F(x) = e^{x} - e^{-1} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1080

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} 3 t^{2} - 2 t \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \left(t^{3} - t^{2}\right) = 3 t^{2} - 2 t \]
    An antiderivative of the integrand is t**3 - t**2.✓ Proved
  2. \[ - \left. t^{3} - t^{2} \right|_{\substack{ t=0 }} + \left. t^{3} - t^{2} \right|_{\substack{ t=x }} = x^{2} \left(x - 1\right) \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} x^{2} \left(x - 1\right) = 3 x^{2} - 2 x \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. x^{2} \left(x - 1\right) \right|_{\substack{ x=0 }} = 0 \]
    And F(0) = 0.✓ Proved
Answer \( F(x) = x^{2} \left(x - 1\right) \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1081

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t^{2} \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \frac{t^{3}}{3} = t^{2} \]
    An antiderivative of the integrand is t**3/3.✓ Proved
  2. \[ - \left. \frac{t^{3}}{3} \right|_{\substack{ t=0 }} + \left. \frac{t^{3}}{3} \right|_{\substack{ t=x }} = \frac{x^{3}}{3} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \frac{x^{3}}{3} = x^{2} \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \frac{x^{3}}{3} \right|_{\substack{ x=0 }} = 0 \]
    And F(0) = 0.✓ Proved
Answer \( F(x) = \frac{x^{3}}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1082

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} \sqrt{t} \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \frac{2 t^{\frac{3}{2}}}{3} = \sqrt{t} \]
    An antiderivative of the integrand is 2*t**(3/2)/3.✓ Proved
  2. \[ - \left. \frac{2 t^{\frac{3}{2}}}{3} \right|_{\substack{ t=1 }} + \left. \frac{2 t^{\frac{3}{2}}}{3} \right|_{\substack{ t=x }} = \frac{2 x^{\frac{3}{2}}}{3} - \frac{2}{3} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(\frac{2 x^{\frac{3}{2}}}{3} - \frac{2}{3}\right) = \sqrt{x} \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \frac{2 x^{\frac{3}{2}}}{3} - \frac{2}{3} \right|_{\substack{ x=1 }} = 0 \]
    And F(1) = 0.✓ Proved
Answer \( F(x) = \frac{2 x^{\frac{3}{2}}}{3} - \frac{2}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1086

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t^{2} \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \frac{t^{3}}{3} = t^{2} \]
    An antiderivative of the integrand is t**3/3.✓ Proved
  2. \[ - \left. \frac{t^{3}}{3} \right|_{\substack{ t=-1 }} + \left. \frac{t^{3}}{3} \right|_{\substack{ t=x }} = \frac{x^{3}}{3} + \frac{1}{3} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(\frac{x^{3}}{3} + \frac{1}{3}\right) = x^{2} \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \frac{x^{3}}{3} + 1 \cdot \frac{1}{3} \right|_{\substack{ x=-1 }} = 0 \]
    And F(-1) = 0.✓ Proved
Answer \( F(x) = \frac{x^{3}}{3} + \frac{1}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1088

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t - 3 \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \left(\frac{t^{2}}{2} - 3 t\right) = t - 3 \]
    An antiderivative of the integrand is t**2/2 - 3*t.✓ Proved
  2. \[ - \left. \frac{t^{2}}{2} - 3 t \right|_{\substack{ t=0 }} + \left. \frac{t^{2}}{2} - 3 t \right|_{\substack{ t=x }} = \frac{x \left(x - 6\right)}{2} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \frac{x \left(x - 6\right)}{2} = x - 3 \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \frac{x \left(x - 6\right)}{2} \right|_{\substack{ x=0 }} = 0 \]
    And F(0) = 0.✓ Proved
Answer \( F(x) = \frac{x \left(x - 6\right)}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1089

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t - 1 \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \left(\frac{t^{2}}{2} - t\right) = t - 1 \]
    An antiderivative of the integrand is t**2/2 - t.✓ Proved
  2. \[ - \left. \frac{t^{2}}{2} - t \right|_{\substack{ t=-1 }} + \left. \frac{t^{2}}{2} - t \right|_{\substack{ t=x }} = \frac{x^{2}}{2} - x - \frac{3}{2} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - x - \frac{3}{2}\right) = x - 1 \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \frac{x^{2}}{2} - x - \frac{3}{2} \right|_{\substack{ x=-1 }} = 0 \]
    And F(-1) = 0.✓ Proved
Answer \( F(x) = \frac{x^{2}}{2} - x - \frac{3}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1090

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} \cos{\left(t \right)} \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]
    An antiderivative of the integrand is sin(t).✓ Proved
  2. \[ - \left. \sin{\left(t \right)} \right|_{\substack{ t=0 }} + \left. \sin{\left(t \right)} \right|_{\substack{ t=x }} = \sin{\left(x \right)} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)} \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]
    And F(0) = 0.✓ Proved
Answer \( F(x) = \sin{\left(x \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1134

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t + 4 \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \left(\frac{t^{2}}{2} + 4 t\right) = t + 4 \]
    An antiderivative of the integrand is t**2/2 + 4*t.✓ Proved
  2. \[ - \left. \frac{t^{2}}{2} + 4 t \right|_{\substack{ t=0 }} + \left. \frac{t^{2}}{2} + 4 t \right|_{\substack{ t=x }} = \frac{x \left(x + 8\right)}{2} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \frac{x \left(x + 8\right)}{2} = x + 4 \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \frac{x \left(x + 8\right)}{2} \right|_{\substack{ x=0 }} = 0 \]
    And F(0) = 0.✓ Proved
Answer \( F(x) = \frac{x \left(x + 8\right)}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1138

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} \cos{\left(t \right)} \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]
    An antiderivative of the integrand is sin(t).✓ Proved
  2. \[ - \left. \sin{\left(t \right)} \right|_{\substack{ t=-1 }} + \left. \sin{\left(t \right)} \right|_{\substack{ t=x }} = \sin{\left(x \right)} + \sin{\left(1 \right)} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(\sin{\left(x \right)} + \sin{\left(1 \right)}\right) = \cos{\left(x \right)} \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \sin{\left(x \right)} + \sin{\left(1 \right)} \right|_{\substack{ x=-1 }} = 0 \]
    And F(-1) = 0.✓ Proved
Answer \( F(x) = \sin{\left(x \right)} + \sin{\left(1 \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1198

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} 3 t^{2} - 2 t \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \left(t^{3} - t^{2}\right) = 3 t^{2} - 2 t \]
    An antiderivative of the integrand is t**3 - t**2.✓ Proved
  2. \[ - \left. t^{3} - t^{2} \right|_{\substack{ t=-1 }} + \left. t^{3} - t^{2} \right|_{\substack{ t=x }} = x^{3} - x^{2} + 2 \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(x^{3} - x^{2} + 2\right) = 3 x^{2} - 2 x \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. x^{3} - x^{2} + 2 \right|_{\substack{ x=-1 }} = 0 \]
    And F(-1) = 0.✓ Proved
Answer \( F(x) = x^{3} - x^{2} + 2 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1202

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} 2 t \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} t^{2} = 2 t \]
    An antiderivative of the integrand is t**2.✓ Proved
  2. \[ - \left. t^{2} \right|_{\substack{ t=-1 }} + \left. t^{2} \right|_{\substack{ t=x }} = x^{2} - 1 \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(x^{2} - 1\right) = 2 x \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. x^{2} - 1 \right|_{\substack{ x=-1 }} = 0 \]
    And F(-1) = 0.✓ Proved
Answer \( F(x) = x^{2} - 1 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1254

easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t + 1 \, dt \) as a function of \( \displaystyle x \).
  1. \[ \frac{d}{d t} \left(\frac{t^{2}}{2} + t\right) = t + 1 \]
    An antiderivative of the integrand is t**2/2 + t.✓ Proved
  2. \[ - \left. \frac{t^{2}}{2} + t \right|_{\substack{ t=-1 }} + \left. \frac{t^{2}}{2} + t \right|_{\substack{ t=x }} = \frac{x^{2}}{2} + x + \frac{1}{2} \]
    F(x) = G(x) − G(a).✓ Proved
  3. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) = x + 1 \]
    F′(x) = f(x), as FTC Part 1 says.✓ Proved
  4. \[ \left. \frac{x^{2}}{2} + x + 1 \cdot \frac{1}{2} \right|_{\substack{ x=-1 }} = 0 \]
    And F(-1) = 0.✓ Proved
Answer \( F(x) = \frac{x^{2}}{2} + x + \frac{1}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More accumulation functions practice

Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} e^{t} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1078easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} \frac{1}{t} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1083easy✓ Every equation proved
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} 3 t^{2} - 2 t \, dt \) as a function of \( \displaystyle x \).
Problem 4.1084easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} \sqrt{t} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1085easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t^{3} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1087easy✓ Every equation proved
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} 3 t^{2} - 2 t \, dt \) as a function of \( \displaystyle x \).
Problem 4.1091easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} \frac{1}{t} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1130easy✓ Every equation proved
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} \cos{\left(t \right)} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1131easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} \cos{\left(t \right)} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1132easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} \sqrt{t} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1133easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t + 4 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1135easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t - 4 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1136easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t^{2} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1137easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t + 3 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1139easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t - 1 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1197easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t^{2} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1199easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t^{3} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1200easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t - 2 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1201easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t + 2 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1203easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} e^{t} \, dt \) as a function of \( \displaystyle x \).
Problem 4.1204easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t - 1 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1205easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t - 2 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1206easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t - 2 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1207easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t - 4 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1251easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t + 4 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1252easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t - 3 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1253easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t - 4 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1255easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t + 1 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1256easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t + 3 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1257easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t - 3 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1258easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t + 3 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1259easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t + 1 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1260easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t + 2 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1313easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t + 2 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1314easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t - 1 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1315easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} t + 1 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1316easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} t + 3 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1388easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t - 2 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1494easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t - 4 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1495easy✓ Nihil obstat
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{-1}^{x} t + 4 \, dt \) as a function of \( \displaystyle x \).
Problem 4.1498easy✓ Nihil obstat