Math / Calculus

QuestionEvaluate the indefinite integral. $\int x(4 x+5)^{8} d x$

Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of $x(4x+5)^8$ , which means finding a function whose derivative is $x(4x+5)^8$ . Watch out! Don't forget the constant of integration since we're dealing with an indefinite integral!
Also, remember the power rule and chain rule for derivatives will be helpful in reverse here.

STEP 2

1. U-Substitution
2. Expand and Simplify
3. Integrate
4. Back-Substitute

STEP 3

Let's begin by making a clever substitution!
We'll let $u = 4x + 5$ .
This is our u-substitution, and it's going to make things much easier!

STEP 4

Now, we need to find $dx$ in terms of $du$ .
Since $u = 4x + 5$ , we can differentiate both sides with respect to $x$ to get $\frac{du}{dx} = 4$ .
This means $du = 4 \cdot dx$ , so $dx = \frac{1}{4} \cdot du$ .
Awesome!

STEP 5

We also need to express $x$ in terms of $u$ .
Since $u = 4x + 5$ , we can solve for $x$ : $4x = u - 5$ , so $x = \frac{u-5}{4}$ .
Perfect!

STEP 6

Now, we can rewrite our integral entirely in terms of $u$ : $\int x(4x+5)^8 dx = \int \frac{u-5}{4} \cdot u^8 \cdot \frac{1}{4} du = \int \frac{u^9 - 5u^8}{16} du$

STEP 7

Let's break up that fraction inside the integral to make it even easier to work with: $\int \frac{u^9 - 5u^8}{16} du = \frac{1}{16} \int (u^9 - 5u^8) du = \frac{1}{16} \int u^9 du - \frac{5}{16} \int u^8 du$

STEP 8

Now, we can integrate each term using the power rule!
Remember, the power rule for integration says $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , where $C$ is the constant of integration.

STEP 9

Applying the power rule, we get: $\frac{1}{16} \int u^9 du - \frac{5}{16} \int u^8 du = \frac{1}{16} \cdot \frac{u^{10}}{10} - \frac{5}{16} \cdot \frac{u^9}{9} + C$ $= \frac{u^{10}}{160} - \frac{5u^9}{144} + C$

STEP 10

Finally, let's substitute back $u = 4x + 5$ to get our final answer in terms of $x$ : $\frac{(4x+5)^{10}}{160} - \frac{5(4x+5)^9}{144} + C$

STEP 11

The indefinite integral of $x(4x+5)^8$ is $\frac{(4x+5)^{10}}{160} - \frac{5(4x+5)^9}{144} + C$ .

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