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Popular >

integral of (x+e^x)x

Solution

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

Solution

\frac{x ^{3}}{3} + e^{x} x - e^{x} + C

Solution steps

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

Expand

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

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

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

\int e^{x} x d x = e^{x} x - e^{x}

= \frac{x ^{3}}{3} + e^{x} x - e^{x}

Add a constant to the solution

= \frac{x ^{3}}{3} + e^{x} x - e^{x} + C

Popular Examples

integral of 1/(4+25x^2)

\int \frac{1}{4 + 25 x ^{2}} d x

integral of (xcos(x))/(sin^3(x))

\int \frac{x cos ( x )}{sin ^{3} ( x )} d x

integral of 3/(x^2+3x)

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

integral of t^2sin^3(t^3)cos^2(t^3)

\int t^{2} sin^{3} (t^{3}) cos^{2} (t^{3}) d t

integral of (-x^2-2)/((x^2+x+1)^2)

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

Frequently Asked Questions (FAQ)

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