QuestionCalculate the integral: $\int x(2 x+5)^{8} d x$

Studdy Solution

STEP 1

Assumptions1. We are asked to compute the integral of a function of the form $x(x+5)^8$ . . We are going to use the method of substitution to solve this integral.

STEP 2

Let's choose a substitution that will simplify the integral. A good choice is to let $u$ equal the inner function of the composite function.
$u =2x +5$

STEP 3

Now we need to find $du$ , the derivative of $u$ with respect to $x$ . This will allow us to substitute $dx$ in the integral.
$du =2 dx$

STEP 4

We see that $dx$ is not alone, but we can solve for it by dividing both sides of the equation by2.
$dx = \frac{1}{2} du$

STEP 5

Now, we substitute $u$ and $dx$ into the integral. Note that $x$ can be expressed in terms of $u$ as $x = \frac{u -5}{2}$ .
$\int x(2 x+5)^{8} d x = \int \left(\frac{u -5}{2}\right) u^8 \left(\frac{1}{2}\right) du$

STEP 6

implify the integral.
$\int \left(\frac{u -5}{2}\right) u^8 \left(\frac{1}{2}\right) du = \frac{1}{4} \int (u^9 -5u^8) du$

STEP 7

Now, we can integrate term by term.
$\frac{1}{4} \int (u^9 -5u^) du = \frac{1}{4} \left(\frac{u^{10}}{10} - \frac{5u^9}{9}\right) + C$

STEP 8

Substitute $u =2x +5$ back into the equation.
\frac{1}{4} \left(\frac{u^{10}}{10} - \frac{5u^}{}\right) + C = \frac{1}{4} \left(\frac{(2x +5)^{10}}{10} - \frac{5(2x +5)^}{}\right) + C

STEP 9

implify the expression to get the final answer.
$\frac{}{4} \left(\frac{(2x +5)^{}}{} - \frac{5(2x +5)^9}{9}\right) + C = \frac{(2x +5)^{}}{40} - \frac{5(2x +5)^9}{36} + C$ So, the integral of $x(2x+5)^8$ with respect to $x$ is $\frac{(2x +5)^{}}{40} - \frac{5(2x +5)^9}{36} + C$ .

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