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

The integral of 1/(sqrt(x(1+x))) is 2ln|sqrt(x+1)+sqrt(x)|+C

Popular >

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

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

2 ln x + 1 + x + C

Solution steps

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x (1 + x) = x (1 + x)

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

Apply u-substitution

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

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

Use the common integral:

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

= 2 ln u^{2} + 1 + u

Substitute back

u = x

= 2 ln (x)^{2} + 1 + x

(x)^{2} = x

= 2 ln x + 1 + x

Add a constant to the solution

= 2 ln x + 1 + x + C

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

\int \frac{1}{sin ^{3} ( x ) cos ( x )} d x

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

\int \frac{x ^{2} + 2 x + 4}{( x - 3 ) ^{5}} d x

What is the integral of 1/(sqrt(x(1+x))) ?
The integral of 1/(sqrt(x(1+x))) is 2ln|sqrt(x+1)+sqrt(x)|+C