Solve \int -2/3 x e^(2x) dx

Evaluate

\int - \frac{2}{3} x e^{2 x} d x

Use the property of integral \int k f (x) d x = k \int f (x) d x

- \frac{2}{3} \times \int x e^{2 x} d x

Prepare for integration by parts

u = x d v = e^{2 x} d x

Calculate the derivative

More Steps

d u = d x d v = e^{2 x} d x

Evaluate the integral

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d u = d x v = \frac{e ^{2 x}}{2}

Substitute u = x 、 v = \frac{e ^{2 x}}{2} 、 d u = d x 、 d v = e^{2 x} d x for \int u d v = uv - \int v d u

- \frac{2}{3} (x \times \frac{e ^{2 x}}{2} - \int 1 \times \frac{e ^{2 x}}{2} d x)

Calculate

- \frac{2}{3} (\frac{x e ^{2 x}}{2} - \int \frac{e ^{2 x}}{2} d x)

Calculate

- \frac{2}{3} \times \frac{x e ^{2 x}}{2} - (- \frac{2}{3} \times \int \frac{e ^{2 x}}{2} d x)

Evaluate

- \frac{2}{3} \times \frac{x e ^{2 x}}{2} + \frac{2}{3} \times \int \frac{e ^{2 x}}{2} d x

Evaluate the integral

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- \frac{2}{3} \times \frac{x e ^{2 x}}{2} + \frac{e ^{2 x}}{6}

Calculate

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- \frac{x e ^{2 x}}{3} + \frac{e ^{2 x}}{6}

Simplify the expression

- \frac{e ^{2 x} x}{3} + \frac{e ^{2 x}}{6}

Solution

- \frac{e ^{2 x} x}{3} + \frac{e ^{2 x}}{6} + C, C \in R