Question

use the substitution ( u = x^{6}+3 ) to find the following indefinite integral. check your answer by differentiating.
int 6 x^{5}left(x^{6}+3
ight)^{7} d x
int 6 x^{5}left(x^{6}+3
ight)^{7} d x=square

Explanation:

Step1: Apply substitution \( u = x^6 + 3 \)

First, find the derivative of \( u \) with respect to \( x \). Using the power rule, \( \frac{du}{dx} = 6x^5 \). Then, we can rewrite this as \( du = 6x^5 dx \).
Looking at the integral \( \int 6x^5 (x^6 + 3)^7 dx \), we see that \( 6x^5 dx \) is exactly \( du \) and \( (x^6 + 3) \) is \( u \). So the integral becomes \( \int u^7 du \).

Step2: Integrate \( \int u^7 du \)

Using the power rule for integration, which states that \( \int u^n du=\frac{u^{n + 1}}{n+1}+C \) (where \( n
eq - 1 \) and \( C \) is the constant of integration), for \( n = 7 \), we have:
\( \int u^7 du=\frac{u^{7 + 1}}{7+1}+C=\frac{u^8}{8}+C \)

Step3: Substitute back \( u = x^6 + 3 \)

Replace \( u \) with \( x^6 + 3 \) in the expression \( \frac{u^8}{8}+C \). We get \( \frac{(x^6 + 3)^8}{8}+C \)

Answer:

\( \frac{(x^6 + 3)^8}{8}+C \) (where \( C \) is the constant of integration)