What is the integral of 5/(sqrt(x)(1+5\sqrt{x))^2} ?

The integral of 5/(sqrt(x)(1+5\sqrt{x))^2} is 5/18 sqrt(x)+C

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integral of 5/(sqrt(x)(1+5\sqrt{x))^2}

\int \frac{5}{x ( 1 + 5 x ) ^{2}} d x

\frac{5}{18} x + C

Solution steps

\int \frac{5}{x ( 1 + 5 x ) ^{2}} d x

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{x ( 1 + 5 x ) ^{2}} = x^{- \frac{1}{2}} \frac{1}{( 1 + 5 ) ^{2}}

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

Take the constant out:

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

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

Simplify

= 5 \cdot \frac{1}{36} \cdot \int x^{- \frac{1}{2}} d x

Apply the Power Rule

: 2 x^{\frac{1}{2}}

= 5 \cdot \frac{1}{36} \cdot 2 x^{\frac{1}{2}}

Simplify

5 \cdot \frac{1}{36} \cdot 2 x^{\frac{1}{2}} : \frac{5}{18} x

= \frac{5}{18} x

Add a constant to the solution

= \frac{5}{18} x + C

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\int \frac{5}{244} x^{4} d x

What is the integral of 5/(sqrt(x)(1+5\sqrt{x))^2} ?
The integral of 5/(sqrt(x)(1+5\sqrt{x))^2} is 5/18 sqrt(x)+C