What is the integral of 7x^3(x-5)^{-4} ?

The integral of 7x^3(x-5)^{-4} is 7(-(x^3)/(3(x-5)^3)+ln|x-5|-(10)/(x-5)-(25)/(2(x-5)^2))+C

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integral of 7x^3(x-5)^{-4}

\int 7 x^{3} (x - 5)^{- 4} d x

7 (- \frac{x ^{3}}{3 ( x - 5 ) ^{3}} + ln ∣ x - 5 ∣ - \frac{10}{x - 5} - \frac{25}{2 ( x - 5 ) ^{2}}) + C

Solution steps

\int 7 x^{3} (x - 5)^{- 4} d x

Take the constant out:

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

= 7 \cdot \int x^{3} (x - 5)^{- 4} d x

Simplify

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

= 7 \cdot \int \frac{x ^{3}}{( x - 5 ) ^{4}} d x

Apply Integration By Parts

= 7 (- \frac{x ^{3}}{3 ( x - 5 ) ^{3}} - \int - \frac{x ^{2}}{( x - 5 ) ^{3}} d x)

\int - \frac{x ^{2}}{( x - 5 ) ^{3}} d x = - ln ∣ x - 5 ∣ + \frac{10}{x - 5} + \frac{25}{2 ( x - 5 ) ^{2}}

= 7 (- \frac{x ^{3}}{3 ( x - 5 ) ^{3}} - (- ln ∣ x - 5 ∣ + \frac{10}{x - 5} + \frac{25}{2 ( x - 5 ) ^{2}}))

Simplify

= 7 (- \frac{x ^{3}}{3 ( x - 5 ) ^{3}} + ln ∣ x - 5 ∣ - \frac{10}{x - 5} - \frac{25}{2 ( x - 5 ) ^{2}})

Add a constant to the solution

= 7 (- \frac{x ^{3}}{3 ( x - 5 ) ^{3}} + ln ∣ x - 5 ∣ - \frac{10}{x - 5} - \frac{25}{2 ( x - 5 ) ^{2}}) + C

What is the integral of 7x^3(x-5)^{-4} ?
The integral of 7x^3(x-5)^{-4} is 7(-(x^3)/(3(x-5)^3)+ln|x-5|-(10)/(x-5)-(25)/(2(x-5)^2))+C