What is the integral of (4x^3+5x)e^{x^2} ?

The integral of (4x^3+5x)e^{x^2} is 2e^{x^2}x^2+1/2 e^{x^2}+C

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integral of (4x^3+5x)e^{x^2}

\int (4 x^{3} + 5 x) e^{x^{2}} d x

2 e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

Solution steps

\int (4 x^{3} + 5 x) e^{x^{2}} d x

Expand

(4 x^{3} + 5 x) e^{x^{2}} : 4 e^{x^{2}} x^{3} + 5 e^{x^{2}} x

= \int 4 e^{x^{2}} x^{3} + 5 e^{x^{2}} x d x

Apply the Sum Rule:

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

= \int 4 e^{x^{2}} x^{3} d x + \int 5 e^{x^{2}} x d x

\int 4 e^{x^{2}} x^{3} d x = 2 (e^{x^{2}} x^{2} - e^{x^{2}})

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

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

Simplify

2 (e^{x^{2}} x^{2} - e^{x^{2}}) + \frac{5}{2} e^{x^{2}} : 2 e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

= 2 e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

Add a constant to the solution

= 2 e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

What is the integral of (4x^3+5x)e^{x^2} ?
The integral of (4x^3+5x)e^{x^2} is 2e^{x^2}x^2+1/2 e^{x^2}+C