What is the integral of (8x+e^x)sqrt(4x^2+e^x) ?

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

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integral of (8x+e^x)sqrt(4x^2+e^x)

\int (8 x + e^{x}) 4 x^{2} + e^{x} d x

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

Solution steps

\int (8 x + e^{x}) 4 x^{2} + e^{x} d x

Apply u-substitution

= \int u d u

Apply the Power Rule

= \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 4 x^{2} + e^{x}

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

Add a constant to the solution

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

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