What is the integral from 0 to pi/4 of 5e^{sin(x)}cos(x) ?

The integral from 0 to pi/4 of 5e^{sin(x)}cos(x) is 5(e^{1/(sqrt(2))}-1)

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integral from 0 to pi/4 of 5e^{sin(x)}cos(x)

\int_{0}^{\frac{π}{4}} 5 e^{s i n (x)} cos (x) d x

5 (e^{\frac{1}{2}} - 1)

Decimal

5.14057 \dots

Solution steps

\int_{0}^{\frac{π}{4}} 5 e^{s i n (x)} cos (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int_{0}^{\frac{π}{4}} e^{s i n (x)} cos (x) d x

Apply u-substitution

: \int_{0}^{\frac{2}{2}} e^{u} d u

= 5 \cdot \int_{0}^{\frac{2}{2}} e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= 5 [e^{u}]_{0}^{\frac{2}{2}}

\frac{2}{2} = \frac{1}{2}

= 5 [e^{u}]_{0}^{\frac{1}{2}}

Compute the boundaries:

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

= 5 (e^{\frac{1}{2}} - 1)

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What is the integral from 0 to pi/4 of 5e^{sin(x)}cos(x) ?
The integral from 0 to pi/4 of 5e^{sin(x)}cos(x) is 5(e^{1/(sqrt(2))}-1)