What is the integral of (x^2-1)/(e^{x/2)} ?

The integral of (x^2-1)/(e^{x/2)} is 2(-(x^2)/(e^{x/2)}-(4x)/(e^{x/2)}-7/(e^{x/2)})+C

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integral of (x^2-1)/(e^{x/2)}

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

2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}}) + C

Solution steps

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

Apply u-substitution

: \int \frac{2 ( 4 u ^{2} - 1 )}{e ^{u}} d u

= \int \frac{2 ( 4 u ^{2} - 1 )}{e ^{u}} d u

Take the constant out:

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

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

Expand

\frac{4 u ^{2} - 1}{e ^{u}} : \frac{4 u ^{2}}{e ^{u}} - \frac{1}{e ^{u}}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{4 u ^{2}}{e ^{u}} d u - \int \frac{1}{e ^{u}} d u)

\int \frac{4 u ^{2}}{e ^{u}} d u = 4 (- \frac{u ^{2}}{e ^{u}} + 2 (- \frac{u}{e ^{u}} - \frac{1}{e ^{u}}))

\int \frac{1}{e ^{u}} d u = - \frac{1}{e ^{u}}

= 2 (4 (- \frac{u ^{2}}{e ^{u}} + 2 (- \frac{u}{e ^{u}} - \frac{1}{e ^{u}})) - (- \frac{1}{e ^{u}}))

Substitute back

u = \frac{x}{2}

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

Simplify

2 (4 (- \frac{( \frac{x}{2} ) ^{2}}{e ^{\frac{x}{2}}} + 2 (- \frac{\frac{x}{2}}{e ^{\frac{x}{2}}} - \frac{1}{e ^{\frac{x}{2}}})) - (- \frac{1}{e ^{\frac{x}{2}}})) : 2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}})

= 2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}})

Add a constant to the solution

= 2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}}) + C

\frac{d}{d x} (3^{x - in x})

\int \frac{x ^{2}}{49 + x ^{2}} d x

x \to 0 lim (2 x)

\frac{d}{d x} (2 x^{2} + 4 x)

1000 e^{- 0.02 p}

What is the integral of (x^2-1)/(e^{x/2)} ?
The integral of (x^2-1)/(e^{x/2)} is 2(-(x^2)/(e^{x/2)}-(4x)/(e^{x/2)}-7/(e^{x/2)})+C