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Chapter 4: Problem 39

Find the indefinite integral. $$ \int \sin 2 x \cos 2 x d x $$

Short Answer

Expert verified
The indefinite integral of \( \sin 2x \cos 2x \) is \( -2\cos(2x) + C \).

Step by step solution

01

Identify and Apply Trigonometric Identity

Recognize that the function inside the integral is a form of the double angle identity in sine, \( \sin(2u) = 2\sin(u)\cos(u) \). Here, \( u = x \). So we could write the integral as \( \int 2\sin(x)\cos(x) \cdot 2 \,dx \) = \( 2\int \sin(2x) \,dx \).
02

Apply u-substitution

Apply substitution method for integration, which involves replacing a part of the integral by a single variable. Here, let \( u = 2x \). Therefore, \( du = 2 \,dx \). This lets us rewrite the integral in terms of \( u \), we get \( \int \sin(u) \,du \).
03

Calculate the Integral

Next, find the indefinite integral of \( \sin(u) \) which is \( -\cos(u) \).
04

Substitute original variables

Replace \( u \) with the original term \( 2x \). So, the integral becomes \( -\cos(2x) \).
05

Apply constant of integration

Insert the constant of integration to solve indefinitely, thus the final answer is \( -2\cos(2x) + C \).

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