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QuestionEvaluate the integral using substitution: ∫−x(5−8x)5 dx=C\int -x(5-8x)^{5} \, dx = C.

Studdy Solution

STEP 1

Assumptions1. We are given the integral ∫−x(5−8x)5dx\int-x(5-8 x)^{5} d x. We need to use a suitable change of variables (i.e., substitution) to evaluate the integral3. We use CC for the constant of integration

STEP 2

We can see that the integral has a function and its derivative. So, we can use the substitution method to simplify the integral. Let's set u=5−8xu =5 -8x.
u=5−8xu =5 -8x

STEP 3

Now, we need to find the derivative of uu with respect to xx, which we denote as du/dxdu/dx.
dudx=−8\frac{du}{dx} = -8

STEP 4

We can rearrange this to find dxdx, which we will substitute back into the integral.
dx=du−8dx = \frac{du}{-8}

STEP 5

Now, we substitute uu and dxdx into the integral.
∫−xu5du−8\int-xu^{5} \frac{du}{-8}

STEP 6

We can simplify this to−18∫u5du-\frac{1}{8}\int u^{5} du

STEP 7

Now, we can evaluate the integral of u5u^{5} with respect to uu.
−1∫u5du=−1⋅u66-\frac{1}{}\int u^{5} du = -\frac{1}{} \cdot \frac{u^{6}}{6}

STEP 8

implify the expression.
−18⋅u66=−u648-\frac{1}{8} \cdot \frac{u^{6}}{6} = -\frac{u^{6}}{48}

STEP 9

Now, we substitute u=5−8xu =5 -8x back into the equation.
−u648=−(5−8x)648-\frac{u^{6}}{48} = -\frac{(5 -8x)^{6}}{48}

STEP 10

Finally, we add the constant of integration, CC, to our solution.
−(5−8x)648+C-\frac{(5 -8x)^{6}}{48} + CSo, the integral of ∫−x(5−8x)5dx\int-x(5-8 x)^{5} d x is −(5−8x)648+C-\frac{(5 -8x)^{6}}{48} + C.

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