Evaluate the indefinite integral.

$\int \left(\mathrm{cot}x\right)\mathrm{ln}\left(\mathrm{sin}x\right)dx$

Pretoto4o
2021-11-19
Answered

Evaluate the indefinite integral.

$\int \left(\mathrm{cot}x\right)\mathrm{ln}\left(\mathrm{sin}x\right)dx$

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Lounctirough

Answered 2021-11-20
Author has **14** answers

Step 1

we have to evaluate the integral

$\int \left(\mathrm{cot}x\right)\mathrm{ln}\left(\mathrm{sin}x\right)dx$ ...(1)

let$\mathrm{cot}x=z\Rightarrow \mathrm{log}\left(\mathrm{sin}x\right)dx=dz$

Step 2

substituting it in the equation (1) we get

$\int zdz$

integrating it we get

$\left[\frac{{z}^{2}}{2}\right]$

putting the value of z we get

$\left[\frac{{\mathrm{cot}}^{2}x}{2}\right]$

therefore the answer of$\int \left(\mathrm{cot}x\right)\mathrm{ln}\left(\mathrm{sin}x\right)dx\text{}is\text{}\frac{{\mathrm{cot}}^{2}x}{2}$

we have to evaluate the integral

let

Step 2

substituting it in the equation (1) we get

integrating it we get

putting the value of z we get

therefore the answer of

Marlene Broomfield

Answered 2021-11-21
Author has **15** answers

Step 1: Remove parentheses.

$\int \mathrm{cot}x\mathrm{ln}\left(\mathrm{sin}x\right)dx$

Step 2: Use Integration by Parts on$\int \mathrm{cot}x\mathrm{ln}\left(\mathrm{sin}x\right)dx$

Let$u=\mathrm{ln}\left(\mathrm{sin}x\right),dv=\mathrm{cot}x,du=\frac{\mathrm{cos}x}{\mathrm{sin}x}dx,v=\mathrm{ln}\left(\mathrm{sin}x\right)$

Step 3: Substitute the above into$uv-\int vdu$ .

${\mathrm{ln}\left(\mathrm{sin}x\right)}^{2}-\int \frac{\mathrm{ln}\left(\mathrm{sin}x\right)\mathrm{cos}x}{\mathrm{sin}x}dx$

Step 4: Use Integration by Substitution on$\int \frac{\mathrm{ln}\left(\mathrm{sin}x\right)\mathrm{cos}x}{\mathrm{sin}x}dx$ .

$\int \frac{\mathrm{ln}u}{u}du$

Step 6: Use Integration by Substitution.

Let$w=\mathrm{ln}u,dw=\frac{1}{u}du$

Step 7: Using w and dw above, rewrite$\int \frac{\mathrm{ln}u}{u}du$ .

$\int wdw$

Step 8: Use Power Rule:$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$ .

$\frac{{w}^{2}}{2}$

Step 9: Substitute$w=\mathrm{ln}u$ back into the original integral.

$\frac{{\mathrm{ln}u}^{2}}{2}$

Step 10: Substitute$u=\mathrm{sin}x$ back into the original integral.

$\frac{{\mathrm{ln}\left(\mathrm{sin}x\right)}^{2}}{2}$

Step 11: Rewrite the integral with the completed substitution.

$\frac{{\mathrm{ln}\left(\mathrm{sin}x\right)}^{2}}{2}$

Step 12: Add constant.

$\frac{{\mathrm{ln}\left(\mathrm{sin}x\right)}^{2}}{2}+C$

Step 2: Use Integration by Parts on

Let

Step 3: Substitute the above into

Step 4: Use Integration by Substitution on

Step 6: Use Integration by Substitution.

Let

Step 7: Using w and dw above, rewrite

Step 8: Use Power Rule:

Step 9: Substitute

Step 10: Substitute

Step 11: Rewrite the integral with the completed substitution.

Step 12: Add constant.

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