What is the integral of sqrt(2x+1)-xe^{x^2} ?

The integral of sqrt(2x+1)-xe^{x^2} is 1/3 (2x+1)^{3/2}-1/2 e^{x^2}+C

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integral of sqrt(2x+1)-xe^{x^2}

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

\frac{1}{3} (2 x + 1)^{\frac{3}{2}} - \frac{1}{2} e^{x^{2}} + C

Solution steps

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

\int 2 x + 1 d x = \frac{1}{3} (2 x + 1)^{\frac{3}{2}}

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

= \frac{1}{3} (2 x + 1)^{\frac{3}{2}} - \frac{1}{2} e^{x^{2}}

Add a constant to the solution

= \frac{1}{3} (2 x + 1)^{\frac{3}{2}} - \frac{1}{2} e^{x^{2}} + C

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

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

\int \frac{x ^{2} + 2 x + 1}{2} sin (1.3 x) d x

\int_{1}^{e^{3}} x^{4} ln (x) d x

f (x) = ln (x e^{x} + 8)

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