What is the integral of ((x+sqrt(x))^2+8e^{2x}) ?

The integral of ((x+sqrt(x))^2+8e^{2x}) is (x^3)/3+4/5 x^{5/2}+(x^2)/2+4e^{2x}+C

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integral of ((x+sqrt(x))^2+8e^{2x})

\int ((x + x)^{2} + 8 e^{2 x}) d x

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

Solution steps

\int (x + x)^{2} + 8 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 + x)^{2} d x + \int 8 e^{2 x} d x

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

\int 8 e^{2 x} d x = 4 e^{2 x}

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

Add a constant to the solution

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

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What is the integral of ((x+sqrt(x))^2+8e^{2x}) ?
The integral of ((x+sqrt(x))^2+8e^{2x}) is (x^3)/3+4/5 x^{5/2}+(x^2)/2+4e^{2x}+C