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

The integral of 1/(sqrt(t)(1+\sqrt{t))} is 2ln|1+sqrt(t)|+C

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

\int \frac{1}{t ( 1 + t )} d t

2 ln 1 + t + C

Solution steps

\int \frac{1}{t ( 1 + t )} d t

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = 1 + t

= 2 ln 1 + t

Add a constant to the solution

= 2 ln 1 + t + C

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What is the integral of 1/(sqrt(t)(1+\sqrt{t))} ?
The integral of 1/(sqrt(t)(1+\sqrt{t))} is 2ln|1+sqrt(t)|+C