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

The integral from 0 to pi of cos(x)sinh(x) is (-e^{2pi}-1)/(4e^pi)-1/2

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integral from 0 to pi of cos(x)sinh(x)

\int_{0}^{π} cos (x) sinh (x) d x

\frac{- e ^{2 π} - 1}{4 e ^{π}} - \frac{1}{2}

Decimal

- 6.29597 \dots

Solution steps

\int_{0}^{π} cos (x) sinh (x) d x

Apply Integration By Parts

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

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

\int - sin (x) cosh (x) d x = - \frac{1}{4} (- e^{- x} cos (x) - e^{- x} sin (x) - e^{x} cos (x) + e^{x} sin (x))

= [cos (x) cosh (x) - (- \frac{1}{4} (- e^{- x} cos (x) - e^{- x} sin (x) - e^{x} cos (x) + e^{x} sin (x)))]_{0}^{π}

Simplify

[cos (x) cosh (x) - (- \frac{1}{4} (- e^{- x} cos (x) - e^{- x} sin (x) - e^{x} cos (x) + e^{x} sin (x)))]_{0}^{π} : [\frac{1}{4} (4 cos (x) cosh (x) - e^{- x} cos (x) - e^{- x} sin (x) - e^{x} cos (x) + e^{x} sin (x))]_{0}^{π}

= [\frac{1}{4} (4 cos (x) cosh (x) - e^{- x} cos (x) - e^{- x} sin (x) - e^{x} cos (x) + e^{x} sin (x))]_{0}^{π}

Compute the boundaries:

\frac{- e ^{2 π} - 1}{4 e ^{π}} - \frac{1}{2}

= \frac{- e ^{2 π} - 1}{4 e ^{π}} - \frac{1}{2}

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\int_{0}^{4} x e^{x} d x

\int_{0}^{4} 2 x^{2} - 8 x d x

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What is the integral from 0 to pi of cos(x)sinh(x) ?
The integral from 0 to pi of cos(x)sinh(x) is (-e^{2pi}-1)/(4e^pi)-1/2