What is the integral of (15)/(5x+xsqrt(x)) ?

The integral of (15)/(5x+xsqrt(x)) is 30(1/5 ln|sqrt(x)|-1/5 ln|sqrt(x)+5|)+C

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integral of (15)/(5x+xsqrt(x))

\int \frac{15}{5 x + x x} d x

30 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5) + C

Solution steps

\int \frac{15}{5 x + x x} d x

Take the constant out:

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

= 15 \cdot \int \frac{1}{5 x + x x} d x

Apply u-substitution

: \int \frac{2}{u ( u + 5 )} d u

= 15 \cdot \int \frac{2}{u ( u + 5 )} d u

Take the constant out:

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

= 15 \cdot 2 \cdot \int \frac{1}{u ( u + 5 )} d u

Take the partial fraction of

\frac{1}{u ( u + 5 )} : \frac{1}{5 u} - \frac{1}{5 ( u + 5 )}

= 15 \cdot 2 \cdot \int \frac{1}{5 u} - \frac{1}{5 ( u + 5 )} 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

= 15 \cdot 2 (\int \frac{1}{5 u} d u - \int \frac{1}{5 ( u + 5 )} d u)

\int \frac{1}{5 u} d u = \frac{1}{5} ln ∣ u ∣

\int \frac{1}{5 ( u + 5 )} d u = \frac{1}{5} ln ∣ u + 5 ∣

= 15 \cdot 2 (\frac{1}{5} ln ∣ u ∣ - \frac{1}{5} ln ∣ u + 5 ∣)

Substitute back

u = x

= 15 \cdot 2 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5)

Simplify

= 30 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5)

Add a constant to the solution

= 30 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5) + C

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What is the integral of (15)/(5x+xsqrt(x)) ?
The integral of (15)/(5x+xsqrt(x)) is 30(1/5 ln|sqrt(x)|-1/5 ln|sqrt(x)+5|)+C