Question

have something similar when wook at the integrand and think n rule, then the antiderivative v u - substitution. \\( \int 6x^2(x^3 + 4)^5 dx \\) 5. \\( \int \cot( \\)

Explanation:

Step1: Choose substitution

Let \( u = x^3 + 4 \). Then, find \( du \) by differentiating \( u \) with respect to \( x \).
\( \frac{du}{dx} = 3x^2 \), so \( du = 3x^2 dx \), and we can solve for \( dx \) to get \( dx = \frac{du}{3x^2} \). Also, notice that \( 6x^2 dx = 2 \cdot 3x^2 dx = 2 du \).

Step2: Substitute into integral

Substitute \( u = x^3 + 4 \) and \( 6x^2 dx = 2 du \) into the integral \( \int 6x^2(x^3 + 4)^5 dx \).
The integral becomes \( \int 2 u^5 du \).

Step3: Integrate with power rule

Use the power rule for integration, \( \int u^n du=\frac{u^{n + 1}}{n+1}+C \) (where \( n = 5 \) here).
\( \int 2u^5 du=2\times\frac{u^{5 + 1}}{5+1}+C=\frac{2u^6}{6}+C=\frac{u^6}{3}+C \)

Step4: Substitute back \( u \)

Substitute \( u = x^3 + 4 \) back into the expression.
We get \( \frac{(x^3 + 4)^6}{3}+C \)

Answer:

\( \frac{(x^3 + 4)^6}{3}+C \)