What is the integral of (x^2)/((x-5)^3) ?

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

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integral of (x^2)/((x-5)^3)

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

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

Solution steps

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

Apply u-substitution

: \int \frac{( u + 5 ) ^{2}}{u ^{3}} d u

= \int \frac{( u + 5 ) ^{2}}{u ^{3}} d u

Expand

\frac{( u + 5 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{10}{u ^{2}} + \frac{25}{u ^{3}}

= \int \frac{1}{u} + \frac{10}{u ^{2}} + \frac{25}{u ^{3}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u} d u + \int \frac{10}{u ^{2}} d u + \int \frac{25}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{10}{u ^{2}} d u = - \frac{10}{u}

\int \frac{25}{u ^{3}} d u = - \frac{25}{2 u ^{2}}

= ln ∣ u ∣ - \frac{10}{u} - \frac{25}{2 u ^{2}}

Substitute back

u = x - 5

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

Add a constant to the solution

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

\frac{d}{d x} (cos (2 - 7 x))

\frac{x}{5 - x}

f (x) = (2 x - 8)^{4} (x^{2} + x + 1)^{5}

y = x^{2} - 4 x, (5, 5)

\frac{\partial}{\partial x} (x^{2} e^{yz})

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