Question

In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int x \cos \left(2 x^{2}\right) d x, \quad u=2 x^{2}$$

Short answer

The result of the integral \(\int x \cos \left(2 x^{2}\right) d x\) using the substitution \( u = 2x^2 \) is \( (1/4) \sin(2x^2) + C \)

Step-by-step solution

Apply substitution

Given, \( u = 2x^2 \), take the derivative of \( u \) with respect to \( x \) to obtain \( du/dx = 4x \). By rearranging we get, \( dx = du / (4x) \)

Substitute \( x \) and \( dx \) in the original Integral

Substitute \( u \) and \( dx \) into the integral, which changes the integral to: \(\int (1/4) \cos(u) du \).

Compute the new integral

Now, calculate the integral of cosine which is simply sin. Therefore, our integral changes to \((1/4) \sin(u) + C \)

Substitute back \( u \)

Now substitute back \( u= 2x^2 \) to get the final integral result as \((1/4) \sin(2x^2) + C \)

Confirm by differentiation

To confirm that our answer is correct, we differentiate our answer. The derivative of \((1/4) \sin(2x^2) \) using the chain rule is \(x \cos(2x^2) \), which shows that our integral is correct