Math 152: Worksheet 7

Trigonometric Integrals

Learning Problems

These problems should be completed on your own. If you need hints on solving a problem, there are some provided with each problem. Click on the word "hint" to view it and again to hide it. They go in increasing order of helpfulness, with the last hint mostly giving away how to do the problem. Try to work from the earlier hints to the later ones, as this will give you the practice you need to succeed in this class.

Problem 1

Compute $\displaystyle \int \sin^5(x) \cos^3(x)\ dx $

• Hint 1

  Which of the formulas for powers of sine and cosine does this fit into?

• Hint 2

  The power of cosine is odd, so we can pull off one of them to group into the $du$ term and convert the rest into sine terms.

• Hint 3

  The integral you then want to do should be $\displaystyle \int \sin^5(x)(1 - \sin^2(x)) \cos(x)\ dx$

• Hint 4

  Set $u = \sin(x)$ to solve this integral.

• Answer

  $\displaystyle \frac{1}{6}\sin^6(x) - \frac{1}{8}\sin^8(x) + C$

Problem 2

Compute $\displaystyle \int \sin^2(x)\cos^6(x) \ dx$

• Hint 1

  Which of the formulas for powers of sine and cosine does this fit into?

• Hint 2

  This one needs the reduction formulas. You can either convert everything into sine or cosine. Cosine is probably easier and shorter.

• Hint 3

  If you convert to cosine, the integral you need to evaluate is $\int \cos^6(x) - \cos^8(x)\ dx$

• Hint 4

  You'll need to step down from 8 to 6 to 4 until you get to 2, and then you can apply the formula for $\cos^2(x)$ to get the answer.

• Answer

  $\displaystyle - \frac{1}{8} \cos^7(x)\sin(x) + \frac{1}{48} \cos^5(x) \sin(x) + \frac{5}{192} \cos^3(x) \sin(x) + \frac{15}{384}x + \frac{15}{768} \sin(2x) + C $

Problem 3

Compute $\displaystyle \int \tan^2(x) \sec^4(x) dx$

• Hint 1

  How does this fit into the powers of tangent and secant formulas?

• Hint 2

  We can rewrite $\sec^4(x)$ as $(1 + \tan^2(x))\sec^2(x)$ and use this $\sec^2(x)$ as part of $du$

• Hint 3

  The integral you get to here should be $\displaystyle \int (\tan^2(x) + \tan^4(x)) \sec^2(x)\ dx$

• Hint 4

  Set $u = \tan(x)$ and $du = \sec^2(x)\ dx$, then integrate.

• Answer

  $\displaystyle \frac{1}{3}\tan^3(x) + \frac{1}{5}\tan^5(x) + C$

Problem 4

Compute $\displaystyle \int \tan^2(x) \sec^3(x)\ dx$

• Hint 1

  How does this fit into the formulas from earlier?

• Hint 2

  This one is going to need reduction formulas, so convert everything to powers of secant and go from there.

• Hint 3

  The integral you need to compute using reduction formulas is $\int \sec^5(x) - \sec^3(x)\ dx$

• Answer

  $\displaystyle \frac{1}{4} \tan(x)\sec^3(x) - \frac{1}{8} \tan(x) \sec(x) - \frac{1}{8} \ln|\sec(x) + \tan(x)| + C$

Problem 5

Compute $\int_0^1 \sin(4x) \sin(2x)\ dx$

• Hint 1

  This falls into the product with two different frequencies type of problem.

• Hint 2

  What formula can we use here to help us integrate this?

• Hint 3

  We can use that $\sin{A} \sin{B} = \frac{1}{2}\cos{(A-B)} - \frac{1}{2}\cos{(A+B)}$ to get that $\sin(4x)\sin(2x) = \frac{1}{2}\cos(2x) - \frac{1}{2}\cos(6x)$

• Answer

  $\displaystyle \frac{1}{4}\sin(4x) - \frac{1}{12}\sin(6x) + C$

Submission Problems

Problem 1

Compute $\displaystyle \int \csc^4(x) \cot^4(x)\ dx$. Hint: Cotangent and cosecent work like tangent and secant.

Problem 2

Compute $\displaystyle \int \cos^4(x)\ dx$