# Consider the indefinite integral integral x^7 (6 + 7 x^8)^{12} dx. Then the most appropriate...

## Question:

Consider the indefinite integral {eq}\displaystyle \int x^7 (6 + 7 x^8)^{12}\ dx {/eq}.

Then the most appropriate substitution to simplify this integral is {eq}u = \_\_\_\ {/eq}Then dx = f(x)du where f(x) = {eq}\_\_\_\ {/eq}.

After making the substitution we obtain the integral {eq}\displaystyle \int g (u)\ du {/eq} where g(u) = {eq}\_\_\_\ {/eq}.

This last integral is: = {eq}\_\_\_\ {/eq}+C (Leave out constant of integration from your answer.)

After substituting back for u we obtain the following final form of the answer: = {eq}\_\_\_\ {/eq}+C (Leave out constant of integration from your answer.)

## Integration by Substitution

Finding the function that represents the result of an indefinite integral can be difficult. This is because while we have a number of rules that allow us to find the derivative of a function, the rules for finding the integral are much more complicated, and they may not even lead to a result. One possible technique is substitution.

## Answer and Explanation:

This problem leads us through the method for conducting this integral step by step. Thus, let's follow it by filling in the required blanks.

The most appropriate substitution to simplify this integral is {eq}u = 6+7x^8 {/eq}. Then dx = f(x)du where f(x) = {eq}56x^7 {/eq}. There are two reasons why we have chosen this substitution. The first is because this is what's being taken to the 12th power, which is something we can't deal with as it is. We do know the antiderivative of {eq}u^{12} {/eq}, though. The second is because its derivative contains the piece that's being multiplied outside of the parentheses. Thus, making this substitution would greatly simplify this integral.

After making the substitution we obtain the integral {eq}\displaystyle \frac{1}{56} \int g(u) du {/eq} where g(u) = {eq}u^{12} {/eq}. The reason for the coefficient outside of the integral is because the substitution required a coefficient of 56 inside of the integral. Thus, to make the substitution, we need to cancel it outside the integral.

This last integral is: = {eq}\frac{1}{728}u^{13} {/eq}+C (Leave out constant of integration from your answer.) This is found by reversing the power rule for differentiation.

After substituting back for u we obtain the following final form of the answer: = {eq}\frac{1}{728} (6+7x^8)^{13} {/eq}

#### Learn more about this topic: How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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