What is the integral of (1+8x)^{3/2} ?

The integral of (1+8x)^{3/2} is 1/20 (1+8x)^{5/2}+C

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integral of (1+8x)^{3/2}

integral

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

\frac{1}{20} (1 + 8 x)^{\frac{5}{2}} + C

Solution steps

\int (1 + 8 x)^{\frac{3}{2}} d x

Apply u-substitution

: \int \frac{u ^{\frac{3}{2}}}{8} d u

= \int \frac{u ^{\frac{3}{2}}}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{1}{8} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = 1 + 8 x

= \frac{1}{8} \cdot \frac{2}{5} (1 + 8 x)^{\frac{5}{2}}

Simplify

\frac{1}{8} \cdot \frac{2}{5} (1 + 8 x)^{\frac{5}{2}} : \frac{1}{20} (1 + 8 x)^{\frac{5}{2}}

= \frac{1}{20} (1 + 8 x)^{\frac{5}{2}}

Add a constant to the solution

= \frac{1}{20} (1 + 8 x)^{\frac{5}{2}} + C

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

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

\frac{1}{( 5 + 2 x + x ^{2} ) ^{\frac{3}{2}}}

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

cos (x) - 1

What is the integral of (1+8x)^{3/2} ?
The integral of (1+8x)^{3/2} is 1/20 (1+8x)^{5/2}+C