20 practice problems across the topics a Calculus 2 final usually covers, in course order,
each with a worked solution checked line by line. Your own exam may differ, so check your syllabus.
Try each problem on paper before opening its solution. If one gives you trouble, the topic link above
it has more like it. How we verify
Find \( \displaystyle \int \left(2 x - 1\right) \cos{\left(4 x - 2 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(2 x - 1\right) \cos{\left(4 x - 2 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(2 x \cos{\left(4 x - 2 \right)} - \cos{\left(4 x - 2 \right)}\right)\, dx \]
linearityDistribute the cosine term.✓ Proved
\[ = 2 \int x \cos{\left(4 x - 2 \right)}\, dx - \int \cos{\left(4 x - 2 \right)}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{x \sin{\left(4 x - 2 \right)}}{2} - 2 \int \frac{\sin{\left(4 x - 2 \right)}}{4}\, dx - \int \cos{\left(4 x - 2 \right)}\, dx \]
partsalgebrasimplifyApply integration by parts to the first integral, where u = x and dv = cos(4*x - 2)dx. Distribute the 2 into the parentheses. Simplify the coefficient of the first term.✓ Proved
\[ = \frac{x \sin{\left(4 x - 2 \right)}}{2} + \frac{\cos{\left(4 x - 2 \right)}}{8} - \int \cos{\left(4 x - 2 \right)}\, dx \]
antiderivativealgebraIntegrate the sine term. Simplify the sign and fraction.✓ Proved
\[ = \frac{x \sin{\left(4 x - 2 \right)}}{2} - \frac{\sin{\left(4 x - 2 \right)}}{4} + \frac{\cos{\left(4 x - 2 \right)}}{8} \]
antiderivativesimplifyIntegrate the final cosine term. Final simplification of the antiderivative.✓ Proved
Answer \( \frac{x \sin{\left(4 x - 2 \right)}}{2} - \frac{\sin{\left(4 x - 2 \right)}}{4} + \frac{\cos{\left(4 x - 2 \right)}}{8} + C \)
Top: the graph of F = x·sin(4x - 2)/2 - sin(4x - 2)/4 + cos(4x - 2)/8. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x - 1)·cos(4x - 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x - 1\right) \cos{\left(4 x - 2 \right)}\, dx \]
integral✓ Proved
\[ = \frac{\int \left(4 x - 2\right) \cos{\left(4 x - 2 \right)}\, dx}{2} \]
algebraFactor 2*x-1 as (1/2)*(4*x-2)✓ Proved
\[ = \frac{\left(4 x - 2\right) \sin{\left(4 x - 2 \right)}}{8} - \frac{\int \sin{\left(4 x - 2 \right)}\, dx}{2} \]
partsIntegrate by parts with u=4*x-2, dv=cos(4*x-2)dx✓ Proved
\[ = \frac{\left(4 x - 2\right) \sin{\left(4 x - 2 \right)}}{8} + \frac{\cos{\left(4 x - 2 \right)}}{8} \]
antiderivativesimplifyIntegral of sin(4*x-2) is -cos(4*x-2)/4 Combine constants✓ Proved
✓ 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
✓ 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
Differentiating the answer returns the integrand.✓ Proved
Answer \( - \frac{\sqrt{x^{2} + 9}}{9 x} + C \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. 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
Top: the graph of F = -log(x)/4 + 3log(x + 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x - 1)/(2x·(2x + 2)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \frac{2 x - 1}{2 x \left(2 x + 2\right)}\, dx \]
integral✓ Proved
\[ = \int \frac{2 x - 1}{4 x \left(x + 1\right)}\, dx \]
simplifyCombine constants in the denominator.✓ Proved
✓ 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
✓ Nihil obstat Lines: 2 proved, 3 reviewed. 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
✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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
✓ Nihil obstat Lines: 1 proved, 1 reviewed. 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
Find the area of the surface obtained by rotating \( \displaystyle y = 2 x \), \( \displaystyle 0 \le x \le 3 \), about the \( \displaystyle x \)-axis.
\[ \frac{d}{d x} 2 x = 2 \]
y′.✓ Proved
Rotating about the x-axis, the radius is y, so S = ∫ 2π·(radius)·√(1 + (y′)²) dx.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. 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
A force of 30 N holds a spring stretched \( \displaystyle \frac{1}{5} \) m beyond its natural length. How much work does it take to stretch it from \( \displaystyle \frac{2}{5} \) m to \( \displaystyle \frac{1}{2} \) m beyond its natural length?
Hooke's law: F(x) = kx, with x the stretch beyond natural length.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. 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
✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 y,\ y(0) = 4 \).
Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
Reviewed
Use y(0) to find the constant of integration, then solve for y.
Reviewed
\[ \frac{d}{d x} 4 e^{2 x} = 8 e^{2 x} \]
The solution satisfies the equation.✓ Proved
\[ 4 \]
And the initial condition.Reviewed
Answer \( y = 4 e^{2 x} \)
The slope field of y' = 2y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 4) is highlighted, y = 4e^(2x).
✓ Nihil obstat Lines: 1 proved, 3 reviewed. 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
The slope field of y' = -2y + e^(-2x): short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 4) is highlighted, y = (x + 4)·e^(-2x).
✓ Nihil obstat Lines: 3 proved, 3 reviewed. 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
Does \( \displaystyle \sum_{n=0}^{\infty} - \left(- \frac{3}{5}\right)^{n} \) converge? If so, find its sum.
\[ -1 \]
The first term, a.Reviewed
\[ - \frac{3}{5} \]
The common ratio, r.Reviewed
|r| = 3/5 < 1, so the series converges to a/(1 − r).
Reviewed
\[ - \frac{5}{8} \]
a/(1 − r).Reviewed
Answer \( - \frac{5}{8} \)
Bars show the terms -(-3/5)^n for n = 0 to 23; dots show the running totals. The totals level off at -5/8.
✓ Nihil obstat Lines: 4 reviewed. 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
The limit is 0 < 1, so the series converges.✓ Proved
Answer \( \text{converges} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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
The limit is finite and positive, so the two series do the same thing.✓ Proved
Σ 1/n^2 converges (p > 1), so the given series converges.
Reviewed
Answer \( \text{converges} \)
✓ Nihil obstat Lines: 1 proved, 2 reviewed. 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
The series converges when 1/3|x − 1| < 1, that is |x − 1| < 3.
Reviewed
Answer \( R = 3 \)
✓ Nihil obstat Lines: 1 proved, 2 reviewed. 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
✓ Nihil obstat Lines: 7 proved, 1 reviewed. 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
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{3} \) at \( \displaystyle t = 1 \).
dy/dx = (dy/dt)/(dx/dt).
Reviewed
\[ \frac{d}{d t} e^{t} = e^{t} \]
dx/dt.✓ Proved
\[ \frac{d}{d t} t^{3} = 3 t^{2} \]
dy/dt.✓ Proved
\[ \frac{3}{e} \]
The slope at t = 1.Reviewed
\[ \left[\begin{matrix}e\\1\end{matrix}\right] \]
The point.Reviewed
\[ \frac{3 x - 3 e}{e} + 1 = \frac{3 x}{e} - 2 \]
Point-slope form.✓ Proved
Answer \( y = \frac{3 x}{e} - 2 \)
✓ Nihil obstat Lines: 3 proved, 3 reviewed. 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
Integrate the speed (sin(t/2) ≥ 0 on [0, 2π]).✓ Proved
Answer \( 24 \approx 24.00000 \)
✓ 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
✓ Nihil obstat Lines: 1 proved, 2 reviewed. 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