What is the integral of 1/2 xe^{4x} ?

The integral of 1/2 xe^{4x} is 1/32 (4e^{4x}x-e^{4x})+C

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integral of 1/2 xe^{4x}

\int \frac{1}{2} x e^{4 x} d x

\frac{1}{32} (4 e^{4 x} x - e^{4 x}) + C

Solution steps

\int \frac{1}{2} x e^{4 x} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x e^{4 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{16} d u

= \frac{1}{2} \cdot \int \frac{e ^{u} u}{16} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{16} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{1}{2} \cdot \frac{1}{16} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{2} \cdot \frac{1}{16} (e^{u} u - e^{u})

Substitute back

u = 4 x

= \frac{1}{2} \cdot \frac{1}{16} (e^{4 x} \cdot 4 x - e^{4 x})

Simplify

\frac{1}{2} \cdot \frac{1}{16} (e^{4 x} \cdot 4 x - e^{4 x}) : \frac{1}{32} (4 e^{4 x} x - e^{4 x})

= \frac{1}{32} (4 e^{4 x} x - e^{4 x})

Add a constant to the solution

= \frac{1}{32} (4 e^{4 x} x - e^{4 x}) + C

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What is the integral of 1/2 xe^{4x} ?
The integral of 1/2 xe^{4x} is 1/32 (4e^{4x}x-e^{4x})+C