How do you find the antiderivative of $cos (x) ln (sin (x))$ $cos(x) Ln(sin(x))$ ?

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Answer 1 · 2017-08-14T12:55:51

Use substitution and parts.

$I = \int cos x ln (sin x) d x$ $I = int cosxln(sinx) dx$

Let $t = sin x$ $t = sinx$ so that $d t = cos x d x$ $dt = cosx dx$ and

$I = \int ln t d t$ $I = int lnt dt$

This can be integrated using parts

$u = ln t$ $u = lnt$ so $d u = \frac{1}{t} d t$ $du = 1/t dt$
$d v = d t$ $dv = dt$ , so $v = t$ $v = t$

$I = t ln t - \int t \cdot \frac{1}{t} d t$ $I = tlnt-int t * 1/t dt$

$= t ln t - \int d t$ $= tlnt- int dt$

$= t ln t - t + C$ $= tlnt-t+C$

Reverse the substitution to finish with

$\int cos x ln (sin x) d x = sin x ln (sin x) - sin x + C$ $int cosxln(sinx) dx = sinxln(sinx) - sinx+C$

How do you find the antiderivative of cos(x)ln(sin(x))cos(x) Ln(sin(x)) ?