What is the integral of (x+e^x)^2 ?

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

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integral of (x+e^x)^2

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

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

Solution steps

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

Expand

(x + e^{x})^{2} : x^{2} + 2 e^{x} x + e^{2 x}

= \int x^{2} + 2 e^{x} x + e^{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 x^{2} d x + \int 2 e^{x} x d x + \int e^{2 x} d x

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

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

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

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

Add a constant to the solution

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

\int \frac{1}{x ^{2} - 3 x} d x

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

\int 4 csc (x) d x

\frac{4}{( s + 1 ) ^{3}}

x \to 0 + lim (x^{2} \cdot ln (x))

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