What is the integral of sqrt(5+8x) ?

The integral of sqrt(5+8x) is 1/12 (5+8x)^{3/2}+C

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integral of sqrt(5+8x)

\int 5 + 8 x d x

\frac{1}{12} (5 + 8 x)^{\frac{3}{2}} + C

Solution steps

\int 5 + 8 x d x

Apply u-substitution

: \int \frac{u}{8} d u

= \int \frac{u}{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 d u

Apply the Power Rule

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

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

Substitute back

u = 5 + 8 x

= \frac{1}{8} \cdot \frac{2}{3} (5 + 8 x)^{\frac{3}{2}}

Simplify

\frac{1}{8} \cdot \frac{2}{3} (5 + 8 x)^{\frac{3}{2}} : \frac{1}{12} (5 + 8 x)^{\frac{3}{2}}

= \frac{1}{12} (5 + 8 x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{12} (5 + 8 x)^{\frac{3}{2}} + C

3 x - 8 y x^{2} + 1 \frac{d y}{d x} = 0, y (0) = 6

\int \frac{x - 3}{x ^{2} + 4 x + 8} d x

\int_{0}^{1} e^{3 x} d x

y = 11 x^{5} - 2 x^{3} + 6 x - 8

4 y^{^{''}} + 2 y^{^{'}} + 7 y = 0

What is the integral of sqrt(5+8x) ?
The integral of sqrt(5+8x) is 1/12 (5+8x)^{3/2}+C