Solve \int \frac{x^2 x}{2e^x}

Question

Evaluate the integral

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

Evaluate

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

Multiply the terms

More Steps

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

Rewrite the expression

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

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

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

Evaluate the integral

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

Prepare for integration by parts

u = x^{3} d v = e^{- x} d x

Calculate the derivative

More Steps

d u = 3 x^{2} d x d v = e^{- x} d x

Evaluate the integral

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d u = 3 x^{2} d x v = - e^{- x}

Substitute u = x^{3} 、 v = - e^{- x} 、 d u = 3 x^{2} d x 、 d v = e^{- x} d x for \int u d v = uv - \int v d u

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

Calculate

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

Calculate

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

Evaluate the integral

More Steps

- \frac{1}{2} x^{3} e^{- x} - \frac{3}{2} x^{2} e^{- x} - 3 x e^{- x} - 3 e^{- x}

Rewrite the expression

More Steps

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

Rewrite the expression

More Steps

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

Solution

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

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