What is the integral of sqrt(a+bx) ?

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

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integral of sqrt(a+bx)

\int a + b x d x

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

Solution steps

\int a + b x d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = a + b x

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

Simplify

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

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

Add a constant to the solution

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

x \to 0 lim (\frac{1 - cos ( 2 x )}{x})

x^{\frac{9}{2}} e^{x}

\frac{d}{d x} (19 x^{2} + 9)

\int x + 4 d x

x \to 1 lim (tan (\frac{π x}{2}))

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