• :00Days
  • :00Hours
  • :00Mins
  • 00Seconds
A new era for learning is coming soonSign up for free
Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Essential Calculus: Early Transcendentals
Found in: Page 326
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Evaluate the Integral: \(\int {{{\cos }^2}} x\sin 2xdx\)

To evaluate the integral \(\int {{{\cos }^2}} x\sin 2xdx\), we will use the identity \({\cos ^2}x = \frac{{1 + \cos 2x}}{2}\) followed by integration formulas

\(\int {{{\cos }^2}} x\sin 2xdx = \frac{{ - 1}}{4}\cos 2x - \frac{1}{{16}}\cos 4x + c\)

See the step by step solution

Step by Step Solution

Step 1: Using trigonometric double angle formula

Let I = \(\int {{{\cos }^2}} x\sin 2xdx\)

As \({\cos ^2}x = \frac{{1 + \cos 2x}}{2}\)

\(I = \int {(\frac{{1 + \cos 2x}}{2}} )\sin 2xdx\)

Step 2: Multiplying functions and splitting of integrals

\(\begin{aligned}{l}I &= \frac{1}{2}\int {(1 + \cos 2x)\sin 2xdx} \\ &= \frac{1}{2}\int {(\sin 2x + \sin 2x\cos 2x)dx} \end{aligned}\)

Now split the integral:

\(I = \frac{1}{2}\int {\sin 2xdx + \frac{1}{2}} \int {\sin 2x\cos 2xdx} \)

(1 Term) (2 Term)

Step 3: Multiply and divide 2 terms by 2 and use identity \(2\sin 2x\cos 2x - \sin 4x\) 

\(\begin{aligned}{l}I &= \frac{1}{2}\int {\sin 2xdx + \frac{1}{4}} \int {2\sin 2x\cos 2xdx} \\ &= \frac{1}{2}\int {\sin 2xdx + \frac{1}{4}} \int {\sin 4xdx} \\ &= \frac{1}{2}(\frac{{ - 2\cos x}}{2}) + \frac{1}{4}(\frac{{ - \cos 4x}}{4}) + c\\\end{aligned}\)

\(I = \frac{{ - \cos 2x}}{4} - \frac{1}{{16}}\cos 4x + c\)

Hence value of integral:

\(\int {{{\cos }^2}} x\sin 2xdx = \frac{{ - 1}}{4}\cos 2x - \frac{1}{{16}}\cos 4x + c\)

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.