What is the integral of sqrt(ax+b) ?

The integral of sqrt(ax+b) is (2(ax+b)^{3/2})/(3a)+C

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integral of sqrt(ax+b)

\int a x + b d x

\frac{2 ( a x + b ) ^{\frac{3}{2}}}{3 a} + C

Solution steps

\int a x + b d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = a x + b

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

Simplify

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

= \frac{2 ( a x + b ) ^{\frac{3}{2}}}{3 a}

Add a constant to the solution

= \frac{2 ( a x + b ) ^{\frac{3}{2}}}{3 a} + C

\frac{d}{d x} (in (x in x))

\frac{d}{d x} (11 - x + 4 ln (2 x - 1))

x \to 0 lim (\frac{∣ x - 4 ∣}{x - 4})

\frac{d}{d x} (x + y + 1)

f (x) = 2 e^{2 e^{x} + x}

What is the integral of sqrt(ax+b) ?
The integral of sqrt(ax+b) is (2(ax+b)^{3/2})/(3a)+C