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

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

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

integral

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

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

Solution steps

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

Multiply by the conjugate of

x + 1 : \frac{x ^{2} - 1}{x - 1}

= \int (\frac{x ^{2} - 1}{x - 1})^{\frac{1}{2}} d x

Apply u-substitution

: \int (u + 2)^{\frac{1}{2}} d u

= \int (u + 2)^{\frac{1}{2}} d u

Apply u-substitution

: \int v^{\frac{1}{2}} d v

= \int v^{\frac{1}{2}} d v

Apply the Power Rule

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

= \frac{2}{3} v^{\frac{3}{2}}

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

sin (\frac{x}{3})

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

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

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

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

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