What is the integral from 0 to 3 of 1/(sqrt(x)(1+x)) ?

The integral from 0 to 3 of 1/(sqrt(x)(1+x)) is (2pi)/3

Popular >

integral from 0 to 3 of 1/(sqrt(x)(1+x))

\int_{0}^{3} \frac{1}{x ( 1 + x )} d x

\frac{2 π}{3}

Decimal

2.09439 \dots

Solution steps

\int_{0}^{3} \frac{1}{x ( 1 + x )} d x

Apply u-substitution

: \int_{0}^{3} \frac{2}{1 + u ^{2}} d u

= \int_{0}^{3} \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_{0}^{3} \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= 2 [arctan (u)]_{0}^{3}

Compute the boundaries:

\frac{π}{3}

= 2 \cdot \frac{π}{3}

Simplify

= \frac{2 π}{3}

\int_{- 1}^{1} e^{∣ x ∣} d x

\int_{0}^{6} - 2 x^{2} + 12 x d x

\int_{a}^{a^{2}} x^{2} d x

\int_{π}^{\frac{3 π}{2}} 3 cos (x) d x

\int_{- 1}^{4} x + 4 - (x^{2} - 2 x) d x

What is the integral from 0 to 3 of 1/(sqrt(x)(1+x)) ?
The integral from 0 to 3 of 1/(sqrt(x)(1+x)) is (2pi)/3