What is the integral from 0 to pi/3 of sin(x)ln(cos(x)) ?

The integral from 0 to pi/3 of sin(x)ln(cos(x)) is -1/2+1/2 ln(2)

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integral from 0 to pi/3 of sin(x)ln(cos(x))

\int_{0}^{\frac{π}{3}} sin (x) ln (cos (x)) d x

- \frac{1}{2} + \frac{1}{2} ln (2)

Decimal Notation

- 0.15342 \dots

Solution steps

\int_{0}^{\frac{π}{3}} sin (x) ln (cos (x)) d x

Apply Integration By Parts

: [- \int sin (x) d x - cos (x) ln (cos (x))]_{0}^{\frac{π}{3}}

= [- \int sin (x) d x - cos (x) ln (cos (x))]_{0}^{\frac{π}{3}}

\int sin (x) d x = - cos (x)

= [- (- cos (x)) - cos (x) ln (cos (x))]_{0}^{\frac{π}{3}}

Simplify

= [cos (x) - cos (x) ln (cos (x))]_{0}^{\frac{π}{3}}

Compute the boundaries:

- \frac{1}{2} + \frac{1}{2} ln (2)

= - \frac{1}{2} + \frac{1}{2} ln (2)

\int cos^{2} (ln (x)) d x

\frac{\partial}{\partial x} (y + (\frac{1}{\frac{1}{x} + \frac{1}{z}}))

\int \frac{x}{1 + x ^{2} + x} d x

n \to \infty lim (∣ n ∣)

\int e^{\frac{4}{x}} d x

What is the integral from 0 to pi/3 of sin(x)ln(cos(x)) ?
The integral from 0 to pi/3 of sin(x)ln(cos(x)) is -1/2+1/2 ln(2)