Problem 39

Question

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

Step-by-Step Solution

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Answer

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

1Step 1: 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 $.

2Step 2: 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 $.

3Step 3: Calculate the Integral

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

4Step 4: Substitute original variables

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

5Step 5: Apply constant of integration

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

Problem 39