Question

use the substitution ( u = 6x^{2}+1 ) to find the following indefinite integral. check your answer by differentiating ( int 12xcos(6x^{2}+1)dx ) ( int 12xcos(6x^{2}+1)dx=square )

Explanation:

Step1: Apply substitution \( u = 6x^2 + 1 \)

First, find the derivative of \( u \) with respect to \( x \). Differentiating \( u = 6x^2 + 1 \) gives \( \frac{du}{dx} = 12x \), which can be rewritten as \( du = 12x \, dx \).

Step2: Substitute into the integral

The integral \( \int 12x \cos(6x^2 + 1) \, dx \) can be rewritten using the substitution. Since \( u = 6x^2 + 1 \) and \( du = 12x \, dx \), the integral becomes \( \int \cos(u) \, du \).

Step3: Integrate with respect to \( u \)

The integral of \( \cos(u) \) with respect to \( u \) is \( \sin(u) + C \), where \( C \) is the constant of integration.

Step4: Substitute back \( u = 6x^2 + 1 \)

Replacing \( u \) with \( 6x^2 + 1 \), we get \( \sin(6x^2 + 1) + C \).

Answer:

\( \sin(6x^2 + 1) + C \)