What is the integral of 8we^{7w} ?

The integral of 8we^{7w} is 8/49 (7e^{7w}w-e^{7w})+C

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integral of 8we^{7w}

\int 8 w e^{7 w} d w

\frac{8}{49} (7 e^{7 w} w - e^{7 w}) + C

Solution steps

\int 8 w e^{7 w} d w

Take the constant out:

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

= 8 \cdot \int w e^{7 w} d w

Apply u-substitution

: \int \frac{e ^{u} u}{49} d u

= 8 \cdot \int \frac{e ^{u} u}{49} d u

Take the constant out:

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

= 8 \cdot \frac{1}{49} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= 8 \cdot \frac{1}{49} (e^{u} u - \int e^{u} d u)

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

= 8 \cdot \frac{1}{49} (e^{u} u - e^{u})

Substitute back

u = 7 w

= 8 \cdot \frac{1}{49} (e^{7 w} \cdot 7 w - e^{7 w})

Simplify

8 \cdot \frac{1}{49} (e^{7 w} \cdot 7 w - e^{7 w}) : \frac{8}{49} (7 e^{7 w} w - e^{7 w})

= \frac{8}{49} (7 e^{7 w} w - e^{7 w})

Add a constant to the solution

= \frac{8}{49} (7 e^{7 w} w - e^{7 w}) + C

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What is the integral of 8we^{7w} ?
The integral of 8we^{7w} is 8/49 (7e^{7w}w-e^{7w})+C