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QuestionEvaluate the integral using substitution: ∫cos⁡(x)(13−cos⁡(x))8sin⁡(x)dx=\int \cos (x)(13-\cos (x))^{8} \sin (x) d x =

Studdy Solution

STEP 1

Assumptions1. We are given the integral ∫cos⁡(x)(13−cos⁡(x))8sin⁡(x)dx\int \cos (x)(13-\cos (x))^{8} \sin (x) dx. . We are asked to use a suitable change of variables (also known as substitution) to evaluate the integral.

STEP 2

The integrand has the form of a product of functions, one of which is a derivative of another. This suggests that we can use the substitution method to simplify the integral.
Let's choose u=13−cos⁡(x)u =13 - \cos(x). This choice is motivated by the presence of (13−cos⁡(x))8(13 - \cos(x))^8 in the integrand.

STEP 3

Now, we need to find the derivative of uu with respect to xx, which we'll denote as du/dxdu/dx.
dudx=−sin⁡(x)\frac{du}{dx} = -\sin(x)

STEP 4

We can rewrite the derivative du/dxdu/dx in terms of dxdx.
du=−sin⁡(x)dxdu = -\sin(x) dx

STEP 5

From the equation in4, we see that sin⁡(x)dx=−du\sin(x) dx = -du. We can substitute this into the original integral.

STEP 6

Substitute uu and dudu into the integral.
∫cos⁡(x)(13−cos⁡(x))8sin⁡(x)dx=∫u8(−du)\int \cos (x)(13-\cos (x))^{8} \sin (x) dx = \int u^8 (-du)

STEP 7

The negative sign can be taken out of the integral.
∫u(−du)=−∫udu\int u^ (-du) = -\int u^ du

STEP 8

Now, we can evaluate the integral of u8u^8 with respect to uu.
−∫u8du=−1u+C-\int u^8 du = -\frac{1}{}u^ + C

STEP 9

Finally, we substitute back u=13−cos⁡(x)u =13 - \cos(x) to get the answer in terms of xx.
−9u9+C=−9(13−cos⁡(x))9+C-\frac{}{9}u^9 + C = -\frac{}{9}(13 - \cos(x))^9 + CSo, the solution to the integral is −9(13−cos⁡(x))9+C-\frac{}{9}(13 - \cos(x))^9 + C.

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