What is the integral of sqrt(e^{2x)+x}*(e^{2x}+1/2) ?

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

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integral of sqrt(e^{2x)+x}*(e^{2x}+1/2)

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u}{2} d u

= \int \frac{u}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = e^{2 x} + x

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

Simplify

\frac{1}{2} \cdot \frac{2}{3} (e^{2 x} + x)^{\frac{3}{2}} : \frac{1}{3} (e^{2 x} + x)^{\frac{3}{2}}

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

Add a constant to the solution

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

f (x) = x^{x} (ln (x) + 1)

f (x) = x e^{4 x}

\frac{\partial}{\partial y} (sin (x^{2} - y^{2}))

cos (e^{x})

\int_{1}^{\infty} 3^{- x} d x

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