What is the integral from 0 to 1 of sin(x)ln(cos(x)) ?

The integral from 0 to 1 of sin(x)ln(cos(x)) is cos(1)-cos(1)ln(cos(1))-1

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integral from 0 to 1 of sin(x)ln(cos(x))

\int_{0}^{1} sin (x) ln (cos (x)) d x

cos (1) - cos (1) ln (cos (1)) - 1

Decimal Notation

- 0.12707 \dots

Solution steps

\int_{0}^{1} sin (x) ln (cos (x)) d x

Apply Integration By Parts

: [- \int sin (x) d x - cos (x) ln (cos (x))]_{0}^{1}

= [- \int sin (x) d x - cos (x) ln (cos (x))]_{0}^{1}

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

= [- (- cos (x)) - cos (x) ln (cos (x))]_{0}^{1}

Simplify

= [cos (x) - cos (x) ln (cos (x))]_{0}^{1}

Compute the boundaries:

cos (1) - cos (1) ln (cos (1)) - 1

= cos (1) - cos (1) ln (cos (1)) - 1

\int_{0}^{1} \frac{u ^{2}}{u ^{2} + 1} d u

\int_{- 6}^{\infty} \frac{1}{x + 8} d x

\int_{0}^{1} x^{9} \cdot e^{- x} d x

\int_{0}^{1} \frac{x}{1 - x ^{3}} d x

\int_{6}^{100} 1 x^{- 2} d x

What is the integral from 0 to 1 of sin(x)ln(cos(x)) ?
The integral from 0 to 1 of sin(x)ln(cos(x)) is cos(1)-cos(1)ln(cos(1))-1