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

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

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

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

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

Solution steps

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

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 + y

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

Simplify

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

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

Add a constant to the solution

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

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