What is the integral of 8/(sqrt(x)(1+8\sqrt{x))^5} ?

The integral of 8/(sqrt(x)(1+8\sqrt{x))^5} is 16/59049 sqrt(x)+C

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integral of 8/(sqrt(x)(1+8\sqrt{x))^5}

\int \frac{8}{x ( 1 + 8 x ) ^{5}} d x

\frac{16}{59049} x + C

Solution steps

\int \frac{8}{x ( 1 + 8 x ) ^{5}} d x

Take the constant out:

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

= 8 \cdot \int \frac{1}{x ( 1 + 8 x ) ^{5}} d x

Apply exponent rule:

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

\frac{1}{x ( 1 + 8 x ) ^{5}} = x^{- \frac{1}{2}} \frac{1}{( 1 + 8 ) ^{5}}

= 8 \cdot \int x^{- \frac{1}{2}} \frac{1}{( 1 + 8 ) ^{5}} d x

Take the constant out:

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

= 8 \cdot \frac{1}{( 1 + 8 ) ^{5}} \cdot \int x^{- \frac{1}{2}} d x

Simplify

= 8 \cdot \frac{1}{59049} \cdot \int x^{- \frac{1}{2}} d x

Apply the Power Rule

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

= 8 \cdot \frac{1}{59049} \cdot 2 x^{\frac{1}{2}}

Simplify

8 \cdot \frac{1}{59049} \cdot 2 x^{\frac{1}{2}} : \frac{16}{59049} x

= \frac{16}{59049} x

Add a constant to the solution

= \frac{16}{59049} x + C

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What is the integral of 8/(sqrt(x)(1+8\sqrt{x))^5} ?
The integral of 8/(sqrt(x)(1+8\sqrt{x))^5} is 16/59049 sqrt(x)+C