What is the integral from 0 to pi of x^5cos(x) ?

The integral from 0 to pi of x^5cos(x) is -5pi^4+60pi^2-240

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integral from 0 to pi of x^5cos(x)

\int_{0}^{π} x^{5} cos (x) d x

- 5 π^{4} + 60 π^{2} - 240

Decimal Notation

- 134.86919 \dots

Solution steps

\int_{0}^{π} x^{5} cos (x) d x

Apply Integration By Parts

: [x^{5} sin (x) - \int 5 x^{4} sin (x) d x]_{0}^{π}

= [x^{5} sin (x) - \int 5 x^{4} sin (x) d x]_{0}^{π}

\int 5 x^{4} sin (x) d x = 5 (- x^{4} cos (x) + 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))))

= [x^{5} sin (x) - 5 (- x^{4} cos (x) + 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))))]_{0}^{π}

Compute the boundaries:

- 5 π^{4} + 60 π^{2} - 240

= - 5 π^{4} + 60 π^{2} - 240

\frac{d y}{d x} = \frac{x}{y}

y^{^{'}} - \frac{8}{x} y = \frac{y ^{4}}{x ^{13}}

10 x

\int \frac{5}{2 - 4 x} d x

\int 2 x + 3 x^{2} d x

What is the integral from 0 to pi of x^5cos(x) ?
The integral from 0 to pi of x^5cos(x) is -5pi^4+60pi^2-240