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

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

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

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

\frac{2 π}{3}

Decimal

2.09439 \dots

Solution steps

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

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