What is the integral of 1/(sqrt(5x)(1+5x)) ?

The integral of 1/(sqrt(5x)(1+5x)) is 2/5 arctan(sqrt(5)sqrt(x))+C

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integral of 1/(sqrt(5x)(1+5x))

\int \frac{1}{5 x ( 1 + 5 x )} d x

\frac{2}{5} arctan (5 x) + C

Solution steps

\int \frac{1}{5 x ( 1 + 5 x )} d x

5 x = 5 x

= \int \frac{1}{5 x ( 1 + 5 x )} d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{2}{1 + 5 u ^{2}} d u

= \frac{1}{5} \cdot \int \frac{2}{1 + 5 u ^{2}} d u

Take the constant out:

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

= \frac{1}{5} \cdot 2 \cdot \int \frac{1}{1 + 5 u ^{2}} d u

Apply Integral Substitution

= \frac{1}{5} \cdot 2 \cdot \int \frac{1}{5 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{5} \cdot 2 \cdot \frac{1}{5} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

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

= \frac{1}{5} \cdot 2 \cdot \frac{1}{5} arctan (v)

Substitute back

= \frac{1}{5} \cdot 2 \cdot \frac{1}{5} arctan (5 x)

Simplify

\frac{1}{5} \cdot 2 \cdot \frac{1}{5} arctan (5 x) : \frac{2}{5} arctan (5 x)

= \frac{2}{5} arctan (5 x)

Add a constant to the solution

= \frac{2}{5} arctan (5 x) + C

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What is the integral of 1/(sqrt(5x)(1+5x)) ?
The integral of 1/(sqrt(5x)(1+5x)) is 2/5 arctan(sqrt(5)sqrt(x))+C