# u-substitution, indefinite integrals

I've looked on the web for an answer to this question, and could not find an example.

Could you push me towards a proper u substitution for the following integral? Please don't solve the problem just state what you would use as a substitution and why.

$$\int(\sin^{10}x \cdot \cos x)\ dx$$

let $$u=\sin x$$ $$du=\cos{x}\ dx$$

$$\int(\sin^{9}x) du$$

should I use trigonometric identities or is another substitution valid?

This question is in the substitution section of the textbook, so it has to be solved with simple substitution

Thanks.

When you substitute $u=\sin x$ you should carry it through, replacing every occurrence of $\sin x$ with $u.$ That reduces the problem to integrating a simple power.

The substution works because $u=\sin x \implies \frac{du}{dx}= \cos x$

You are the using the equivalent operators $\color {red}{\int}u^{10} \color {red}{ \frac {du}{dx}dx} \equiv \color {red}{\int} u^{10} \color{red}{du}$

Notice how the $\cos x$ is swallowed up.

If you need a spoiler:

$$\int u^{10}du=\frac{u^{11}}{11}+C\\=\frac{\sin^{11}(x)}{11}+C$$

• They said please don't solve the problem... – Justpassingby Feb 1 '16 at 19:32
• @Justpassingby sorry – 3SAT Feb 1 '16 at 19:33

thanks to u/justpassingthrough

$$\int(sin^{10}x*cosx)dx$$ let $$u=sinx$$ $$du=cosx$$

$$\int(sin^{10}x*cosx)dx=\int(u^{10})du$$ $$=\frac{1}{11}u^{11}+C=\frac{1}{11}(sinx)^{11}+C$$

cool