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

The integral of sqrt((1+\sqrt{x))/x} is 4/3 (1+sqrt(x))^{3/2}+C

Popular >

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

\int \frac{1 + x}{x} d x

\frac{4}{3} (1 + x)^{\frac{3}{2}} + C

Solution steps

\int \frac{1 + x}{x} d x

Simplify

\frac{1 + x}{x} : \frac{1 + x}{x}

= \int \frac{1 + x}{x} d x

Apply u-substitution

: \int 2 u d u

= \int 2 u d u

Take the constant out:

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

= 2 \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = 1 + x

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

Simplify

2 \cdot \frac{2}{3} (1 + x)^{\frac{3}{2}} : \frac{4}{3} (1 + x)^{\frac{3}{2}}

= \frac{4}{3} (1 + x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{3} (1 + x)^{\frac{3}{2}} + C

\int \frac{x}{4 x ^{2} + 1} d x

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

\int \frac{1}{( x - 3 ) x ^{2} - 6 x + 6} d x

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

\int \frac{1}{cos ^{4} ( x ) sin ^{2} ( x )} d x

What is the integral of sqrt((1+\sqrt{x))/x} ?
The integral of sqrt((1+\sqrt{x))/x} is 4/3 (1+sqrt(x))^{3/2}+C