What is the integral from 0 to pi/2 of x(sin(x)) ?

The integral from 0 to pi/2 of x(sin(x)) is 1

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integral from 0 to pi/2 of x(sin(x))

\int_{0}^{\frac{π}{2}} x (sin (x)) d x

1

Solution steps

\int_{0}^{\frac{π}{2}} x sin (x) d x

Apply Integration By Parts

: [- x cos (x) - \int - cos (x) d x]_{0}^{\frac{π}{2}}

= [- x cos (x) - \int - cos (x) d x]_{0}^{\frac{π}{2}}

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

= [- x cos (x) - (- sin (x))]_{0}^{\frac{π}{2}}

Simplify

= [- x cos (x) + sin (x)]_{0}^{\frac{π}{2}}

Compute the boundaries:

1

= 1

\int_{0}^{6} (x - 1) d x

\int_{0}^{4} z d z

\int_{1}^{2} \frac{2 x + 5}{x ^{2}} d x

\int_{0}^{a} x^{5} a^{3} - x^{3} d x

\int_{0}^{2} 3 x - \frac{3}{2} d x

What is the integral from 0 to pi/2 of x(sin(x)) ?
The integral from 0 to pi/2 of x(sin(x)) is 1