What is the integral of sqrt(\sqrt{2)x+1} ?

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

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integral of sqrt(\sqrt{2)x+1}

\int 2 x + 1 d x

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

Solution steps

\int 2 x + 1 d x

Apply u-substitution

= \int \frac{u}{2} d u

Simplify

\frac{u}{2} : \frac{u}{2}

= \int \frac{u}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u d u

Apply the Power Rule

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

Substitute back

u = 2 x + 1

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

Simplify

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

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

Add a constant to the solution

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

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