What is the integral of 5x*e^{10x} ?

The integral of 5x*e^{10x} is 1/20 (10e^{10x}x-e^{10x})+C

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integral of 5x*e^{10x}

\int 5 x \cdot e^{10 x} d x

\frac{1}{20} (10 e^{10 x} x - e^{10 x}) + C

Solution steps

\int 5 x e^{10 x} d x

Take the constant out:

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

= 5 \cdot \int x e^{10 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{100} d u

= 5 \cdot \int \frac{e ^{u} u}{100} d u

Take the constant out:

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

= 5 \cdot \frac{1}{100} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 5 \cdot \frac{1}{100} (e^{u} u - \int e^{u} d u)

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

= 5 \cdot \frac{1}{100} (e^{u} u - e^{u})

Substitute back

u = 10 x

= 5 \cdot \frac{1}{100} (e^{10 x} \cdot 10 x - e^{10 x})

Simplify

5 \cdot \frac{1}{100} (e^{10 x} \cdot 10 x - e^{10 x}) : \frac{1}{20} (10 e^{10 x} x - e^{10 x})

= \frac{1}{20} (10 e^{10 x} x - e^{10 x})

Add a constant to the solution

= \frac{1}{20} (10 e^{10 x} x - e^{10 x}) + C

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What is the integral of 5x*e^{10x} ?
The integral of 5x*e^{10x} is 1/20 (10e^{10x}x-e^{10x})+C