What is the integral of 5/2 xe^{3/2 x} ?

The integral of 5/2 xe^{3/2 x} is 5/9 e^{(3x)/2}(3x-2)+C

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integral of 5/2 xe^{3/2 x}

\int \frac{5}{2} x e^{\frac{3}{2} x} d x

\frac{5}{9} e^{\frac{3 x}{2}} (3 x - 2) + C

Solution steps

\int \frac{5}{2} x e^{\frac{3}{2} x} d x

Take the constant out:

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

= \frac{5}{2} \cdot \int x e^{\frac{3}{2} x} d x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{5}{2} \cdot \int x e^{\frac{3 x}{2}} d x

Apply Integration By Parts

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

= \frac{5}{2} (\frac{2}{3} e^{\frac{3 x}{2}} x - \int \frac{2}{3} e^{\frac{3 x}{2}} d x)

\int \frac{2}{3} e^{\frac{3 x}{2}} d x = \frac{4}{9} e^{\frac{3 x}{2}}

= \frac{5}{2} (\frac{2}{3} e^{\frac{3 x}{2}} x - \frac{4}{9} e^{\frac{3 x}{2}})

Simplify

\frac{5}{2} (\frac{2}{3} e^{\frac{3 x}{2}} x - \frac{4}{9} e^{\frac{3 x}{2}}) : \frac{5}{9} e^{\frac{3 x}{2}} (3 x - 2)

= \frac{5}{9} e^{\frac{3 x}{2}} (3 x - 2)

Add a constant to the solution

= \frac{5}{9} e^{\frac{3 x}{2}} (3 x - 2) + C

\int 5 x^{2} + 2 d x

\int \frac{1}{x \cdot x + 1} d x

\int 9 z d z

\int \frac{1}{( x - 1 ) ^{4} x ^{2} - 2 x} d x

What is the integral of 5/2 xe^{3/2 x} ?
The integral of 5/2 xe^{3/2 x} is 5/9 e^{(3x)/2}(3x-2)+C