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

= \int u d u

Apply the Power Rule

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

= \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

\frac{d ^{2}}{d x ^{2}} (3 x^{2} cos (8 x))

x \to 1 lim ((ln (x))^{x - 1})

\frac{d y}{d x} = 9 x^{2} - 12 x

\frac{1}{( s - 1 ) ( s - 2 ) ( s - 3 )}

e^{4 x^{3} - 18 x^{2}}

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