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

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

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

Apply u-substitution

= \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

= 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

2 \frac{d y}{d x} - 4 x y = 8 x, y (0) = 2

\int 6 arctan (\frac{8}{x}) d x

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\int \frac{17}{x ^{3} - 27} d x

x \to 1 - lim (\frac{2 x - 3}{x - 1})

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