What is the integral of x^3+sqrt(7x)-5xe^{-2x^2} ?

The integral of x^3+sqrt(7x)-5xe^{-2x^2} is (x^4)/4+sqrt(7) 2/3 x^{3/2}+5/4 e^{-2x^2}+C

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integral of x^3+sqrt(7x)-5xe^{-2x^2}

\int x^{3} + 7 x - 5 x e^{- 2 x^{2}} d x

\frac{x ^{4}}{4} + 7 \frac{2}{3} x^{\frac{3}{2}} + \frac{5}{4} e^{- 2 x^{2}} + C

Solution steps

\int x^{3} + 7 x - 5 x e^{- 2 x^{2}} 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 x^{3} d x + \int 7 x d x - \int 5 x e^{- 2 x^{2}} d x

\int x^{3} d x = \frac{x ^{4}}{4}

\int 7 x d x = 7 \frac{2}{3} x^{\frac{3}{2}}

\int 5 x e^{- 2 x^{2}} d x = - \frac{5}{4} e^{- 2 x^{2}}

= \frac{x ^{4}}{4} + 7 \frac{2}{3} x^{\frac{3}{2}} - (- \frac{5}{4} e^{- 2 x^{2}})

Simplify

= \frac{x ^{4}}{4} + 7 \frac{2}{3} x^{\frac{3}{2}} + \frac{5}{4} e^{- 2 x^{2}}

Add a constant to the solution

= \frac{x ^{4}}{4} + 7 \frac{2}{3} x^{\frac{3}{2}} + \frac{5}{4} e^{- 2 x^{2}} + C

What is the integral of x^3+sqrt(7x)-5xe^{-2x^2} ?
The integral of x^3+sqrt(7x)-5xe^{-2x^2} is (x^4)/4+sqrt(7) 2/3 x^{3/2}+5/4 e^{-2x^2}+C