Evaluate the indefinite integral. \int (\cot x)\ln (\sin x)dx

Evaluate the indefinite integral.
$\int \left(\mathrm{cot}x\right)\mathrm{ln}\left(\mathrm{sin}x\right)dx$
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Lounctirough
Step 1
we have to evaluate the integral
$\int \left(\mathrm{cot}x\right)\mathrm{ln}\left(\mathrm{sin}x\right)dx$...(1)
let $\mathrm{cot}x=z⇒\mathrm{log}\left(\mathrm{sin}x\right)dx=dz$
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
Not exactly what you’re looking for?
Marlene Broomfield
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.
$\frac{{\mathrm{ln}\left(\mathrm{sin}x\right)}^{2}}{2}+C$