Powers and products of sines and cosines, with identities. 49 problems with worked solutions; in 37 of them every
equation is proved by a computer algebra system.
20 worked problems
Try each one before opening the solution. Every line is labeled with what checked it; 20 of these also have a second solution, written separately and checked the same way. 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
\[ = \int \frac{\sin{\left(2 x - 2 \right)}}{2}\, dx \]
rewriteUse sin(u)cos(u)=½sin(2u)✓ Proved
\[ = - \frac{\cos{\left(2 x - 2 \right)}}{4} \]
antiderivative∫sin(2u)du = -½cos(2u)✓ 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
\[ = \frac{x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{12} \]
antiderivativeIntegrate cos(6*x-2) to sin(6*x-2)/6 and divide by 2✓ 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
trig-identityalgebraUse the product-to-sum identity: sin(A)cos(B) = 0.5(sin(A+B) + sin(A-B)). Simplify the arguments inside the sine functions.✓ Proved
Answer \( - \frac{\cos{\left(x + 1 \right)}}{2} - \frac{\cos{\left(3 x + 3 \right)}}{6} + C \)
Top: the graph of F = -cos(x + 1)/2 - cos(3x + 3)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(2x + 2)·cos(x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
\[ = \frac{x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{12} \]
antiderivativesimplifyintegrate cos 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
\[ = \frac{x}{2} + \frac{\sin{\left(6 x + 2 \right)}}{12} \]
antiderivativeIntegrate the cosine term✓ 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
\[ = \int \frac{\sin{\left(2 x + 2 \right)}}{2}\, dx \]
rewriteUse sin(u)cos(u)=1/2*sin(2u)✓ Proved
\[ = - \frac{\cos{\left(2 x + 2 \right)}}{4} \]
antiderivativeIntegral of sin(kx) is -cos(kx)/k✓ 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
trig-identityalgebraUse the product-to-sum identity: sin(A)cos(B) = 0.5(sin(A+B) + sin(A-B)). Simplify the arguments inside the sine functions.✓ Proved
Answer \( - \frac{\cos{\left(x - 1 \right)}}{2} - \frac{\cos{\left(3 x - 3 \right)}}{6} + C \)
Top: the graph of F = -cos(x - 1)/2 - cos(3x - 3)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(2x - 2)·cos(x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
Find \( \displaystyle \int \sin{\left(6 x - 2 \right)} \cos{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \sin{\left(6 x - 2 \right)} \cos{\left(3 x - 1 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(0.5 \sin{\left(3 x - 1 \right)} + 0.5 \sin{\left(9 x - 3 \right)}\right)\, dx \]
trig-identityalgebraUse the product-to-sum identity: sin(A)cos(B) = 0.5(sin(A+B) + sin(A-B)). Simplify the arguments inside the sine functions.✓ Proved
\[ = \int 0.5 \sin{\left(3 x - 1 \right)}\, dx + \int 0.5 \sin{\left(9 x - 3 \right)}\, dx \]
linearitySplit the integral into two parts using linearity.✓ Proved
\[ = - 0.166666666666667 \cos{\left(3 x - 1 \right)} - 0.0555555555555556 \cos{\left(9 x - 3 \right)} \]
antiderivativeIntegrate each sine term using the rule integral sin(ax+b) dx = -cos(ax+b)/a.✓ Proved
Answer \( - \frac{\cos{\left(3 x - 1 \right)}}{6} - \frac{\cos{\left(9 x - 3 \right)}}{18} + C \)
Top: the graph of F = -cos(3x - 1)/6 - cos(9x - 3)/18. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(6x - 2)·cos(3x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \sin{\left(6 x - 2 \right)} \cos{\left(3 x - 1 \right)}\, dx \]
integral✓ Proved
\[ = \int \left(\frac{\sin{\left(3 x - 1 \right)}}{2} + \frac{\sin{\left(9 x - 3 \right)}}{2}\right)\, dx \]
trig-identityUse sinA cosB = ½[sin(A+B)+sin(A-B)]✓ Proved
\[ = \frac{\int \sin{\left(3 x - 1 \right)}\, dx}{2} + \frac{\int \sin{\left(9 x - 3 \right)}\, dx}{2} \]
linearitySplit the integral✓ Proved
\[ = - \frac{\cos{\left(3 x - 1 \right)}}{6} - \frac{\cos{\left(9 x - 3 \right)}}{18} \]
✓ 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
Find \( \displaystyle \int \sin{\left(6 x + 2 \right)} \cos{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \sin{\left(6 x + 2 \right)} \cos{\left(3 x + 1 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(0.5 \sin{\left(3 x + 1 \right)} + 0.5 \sin{\left(9 x + 3 \right)}\right)\, dx \]
trig-identityUse the product-to-sum identity sin(A)cos(B) = 0.5(sin(A+B) + sin(A-B)).✓ Proved
\[ = 0.5 \int \sin{\left(3 x + 1 \right)}\, dx + 0.5 \int \sin{\left(9 x + 3 \right)}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = - 0.166666666666667 \cos{\left(3 x + 1 \right)} - 0.0555555555555556 \cos{\left(9 x + 3 \right)} \]
antiderivativeIntegrate each sine term using the rule integral sin(ax+b) dx = -cos(ax+b)/a.✓ Proved
\[ = - \frac{\cos{\left(3 x + 1 \right)}}{6} - \frac{\cos{\left(9 x + 3 \right)}}{18} \]
simplifySimplify the coefficients.✓ Proved
Answer \( - \frac{\cos{\left(3 x + 1 \right)}}{6} - \frac{\cos{\left(9 x + 3 \right)}}{18} + C \)
Top: the graph of F = -cos(3x + 1)/6 - cos(9x + 3)/18. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(6x + 2)·cos(3x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \sin{\left(6 x + 2 \right)} \cos{\left(3 x + 1 \right)}\, dx \]
integral✓ Proved
\[ = \int \left(\frac{\sin{\left(3 x + 1 \right)}}{2} + \frac{\sin{\left(9 x + 3 \right)}}{2}\right)\, dx \]
✓ 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
Find \( \displaystyle \int \sin{\left(2 x \right)} \cos{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
\[ \int \sin{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{\sin{\left(4 x \right)}}{2}\, dx \]
trig-identityUse the double angle identity sin(2u) = 2sin(u)cos(u) with u = 2x.✓ Proved
\[ = \frac{\int \sin{\left(4 x \right)}\, dx}{2} \]
linearityFactor out the constant 1/2.✓ Proved
\[ = - \frac{\cos{\left(4 x \right)}}{8} \]
antiderivativeIntegrate sin(4*x) to get -cos(4*x)/4 and multiply by 1/2.✓ Proved
Answer \( \frac{\sin^{2}{\left(2 x \right)}}{4} + C \)
Top: the graph of F = sin(2x)²/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(2x)·cos(2x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \sin{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \]
integral✓ Proved
\[ = \int \frac{\sin{\left(4 x \right)}}{2}\, dx \]
trig-identityUse sin(2x)cos(2x)=½sin(4x)✓ Proved
\[ = - \frac{\cos{\left(4 x \right)}}{8} \]
antiderivative∫sin(4x)dx = -¼cos(4x); multiply by ½ gives -1/8cos(4x)✓ 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
antiderivativesimplifysimplifyIntegrate sin x and sin 3x Combine the terms Multiply numerator and denominator by 3 to combine fractions✓ 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
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} \]
antiderivativeIntegrate 1/2 to get x/2 and cos(2x)/2 to get sin(2x)/4.✓ 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
antiderivativesimplifyIntegrate using the power rule for the sine term. Final expression.✓ Proved
Answer \( \frac{\left(\cos{\left(2 x + 2 \right)} + 5\right) \sin{\left(x + 1 \right)}}{6} + C \)
Top: the graph of F = (cos(2x + 2) + 5)·sin(x + 1)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = cos(x + 1)³.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
\[ = \frac{x}{2} - \int \frac{\cos{\left(4 x \right)}}{2}\, dx \]
antiderivative∫1/2 dx = x/2.✓ Proved
\[ = \frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8} \]
antiderivative∫cos(4x) dx = sin(4x)/4, divide by 2.✓ 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
\[ = \frac{x}{2} + \frac{\sin{\left(2 x + 2 \right)}}{4} \]
antiderivativeintegrate cos(2x+2)✓ 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
✓ 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