What is the integral of (24x^3)/((x^4-5)^5) ?

The integral of (24x^3)/((x^4-5)^5) is -3/(2(x^4-5)^4)+C

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integral of (24x^3)/((x^4-5)^5)

\int \frac{24 x ^{3}}{( x ^{4} - 5 ) ^{5}} d x

- \frac{3}{2 ( x ^{4} - 5 ) ^{4}} + C

Solution steps

\int \frac{24 x ^{3}}{( x ^{4} - 5 ) ^{5}} d x

Take the constant out:

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

= 24 \cdot \int \frac{x ^{3}}{( x ^{4} - 5 ) ^{5}} d x

Apply u-substitution

: \int \frac{1}{4 u ^{5}} d u

= 24 \cdot \int \frac{1}{4 u ^{5}} d u

Take the constant out:

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

= 24 \cdot \frac{1}{4} \cdot \int \frac{1}{u ^{5}} d u

Apply the Power Rule

: - \frac{1}{4 u ^{4}}

= 24 \cdot \frac{1}{4} (- \frac{1}{4 u ^{4}})

Substitute back

u = x^{4} - 5

= 24 \cdot \frac{1}{4} (- \frac{1}{4 ( x ^{4} - 5 ) ^{4}})

Simplify

24 \cdot \frac{1}{4} (- \frac{1}{4 ( x ^{4} - 5 ) ^{4}}) : - \frac{3}{2 ( x ^{4} - 5 ) ^{4}}

= - \frac{3}{2 ( x ^{4} - 5 ) ^{4}}

Add a constant to the solution

= - \frac{3}{2 ( x ^{4} - 5 ) ^{4}} + C

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\int e^{7 r - 1} (2 r^{2} - r) d r

What is the integral of (24x^3)/((x^4-5)^5) ?
The integral of (24x^3)/((x^4-5)^5) is -3/(2(x^4-5)^4)+C