Math  /  Calculus

QuestionNext, we integrate the resulting integral with respect to uu using the Power Rule. (Use CC for the constant of integration.) 1u2du=u2du=\begin{aligned} \int \frac{1}{u^{2}} d u & =\int u^{-2} d u \\ & =\square \end{aligned}
Finally, replace uu by g(x)=x7+7xg(x)=x^{7}+7 x to obtain the solution as a function of xx. (Use CC for the constant of integration.) \square

Studdy Solution

STEP 1

1. We are given an integral in terms of u u and need to integrate using the Power Rule.
2. After integration, we will substitute back the expression for u u in terms of x x .

STEP 2

1. Integrate the function u2du \int u^{-2} \, du using the Power Rule.
2. Substitute u=g(x)=x7+7x u = g(x) = x^7 + 7x back into the integrated function.

STEP 3

To integrate u2du \int u^{-2} \, du using the Power Rule, recall that the Power Rule for integration states:
undu=un+1n+1+C \int u^n \, du = \frac{u^{n+1}}{n+1} + C
where n1 n \neq -1 . In this case, n=2 n = -2 .
u2du=u2+12+1+C \int u^{-2} \, du = \frac{u^{-2+1}}{-2+1} + C =u11+C = \frac{u^{-1}}{-1} + C =u1+C = -u^{-1} + C =1u+C = -\frac{1}{u} + C

STEP 4

Now, replace u u with g(x)=x7+7x g(x) = x^7 + 7x in the integrated function:
1u+C=1x7+7x+C -\frac{1}{u} + C = -\frac{1}{x^7 + 7x} + C
The solution as a function of x x is:
1x7+7x+C \boxed{-\frac{1}{x^7 + 7x} + C}

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