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

The integral of (sqrt(t))/(1+t) is 2(-arctan(sqrt(t))+sqrt(t))+C

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integral of (sqrt(t))/(1+t)

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

2 (- arctan (t) + t) + C

Solution steps

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

Apply u-substitution

= \int \frac{2 u ^{2}}{1 + u ^{2}} d u

Take the constant out:

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

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

\frac{u ^{2}}{1 + u ^{2}} = - \frac{1}{1 + u ^{2}} + 1

= 2 \cdot \int - \frac{1}{1 + u ^{2}} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

\int 1 d u = u

= 2 (- arctan (u) + u)

Substitute back

u = t

= 2 (- arctan (t) + t)

Add a constant to the solution

= 2 (- arctan (t) + t) + C

What is the integral of (sqrt(t))/(1+t) ?
The integral of (sqrt(t))/(1+t) is 2(-arctan(sqrt(t))+sqrt(t))+C