What is the integral of (4+16e^8x)^{1/2} ?

The integral of (4+16e^8x)^{1/2} is 1/(3e^8)(1+4e^8x)^{3/2}+C

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integral of (4+16e^8x)^{1/2}

integral

(4 + 16 e^{8} x)^{\frac{1}{2}}

\frac{1}{3 e ^{8}} (1 + 4 e^{8} x)^{\frac{3}{2}} + C

Solution steps

\int (4 + 16 e^{8} x)^{\frac{1}{2}} d x

Apply u-substitution

: \int \frac{u ^{\frac{1}{2}}}{16 e ^{8}} d u

= \int \frac{u ^{\frac{1}{2}}}{16 e ^{8}} d u

Take the constant out:

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

= \frac{1}{16 e ^{8}} \cdot \int u^{\frac{1}{2}} d u

Apply the Power Rule

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

= \frac{1}{16 e ^{8}} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 4 + 16 e^{8} x

= \frac{1}{16 e ^{8}} \cdot \frac{2}{3} (4 + 16 e^{8} x)^{\frac{3}{2}}

Simplify

\frac{1}{16 e ^{8}} \cdot \frac{2}{3} (4 + 16 e^{8} x)^{\frac{3}{2}} : \frac{1}{3 e ^{8}} (1 + 4 e^{8} x)^{\frac{3}{2}}

= \frac{1}{3 e ^{8}} (1 + 4 e^{8} x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3 e ^{8}} (1 + 4 e^{8} x)^{\frac{3}{2}} + C

x^{3} (1 + x^{4})^{3}

\frac{( 1 - x )}{x ^{2}}

\frac{x}{( 2 - x )}

\frac{e ^{- 1}}{x}

(x - 1)^{2}

What is the integral of (4+16e^8x)^{1/2} ?
The integral of (4+16e^8x)^{1/2} is 1/(3e^8)(1+4e^8x)^{3/2}+C