What is the integral of (x+1)/((x-3)^3) ?

The integral of (x+1)/((x-3)^3) is -1/(x-3)-2/((x-3)^2)+C

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integral of (x+1)/((x-3)^3)

\int \frac{x + 1}{( x - 3 ) ^{3}} d x

- \frac{1}{x - 3} - \frac{2}{( x - 3 ) ^{2}} + C

Solution steps

\int \frac{x + 1}{( x - 3 ) ^{3}} d x

Expand

\frac{x + 1}{( x - 3 ) ^{3}} : \frac{x}{( x - 3 ) ^{3}} + \frac{1}{( x - 3 ) ^{3}}

= \int \frac{x}{( x - 3 ) ^{3}} + \frac{1}{( x - 3 ) ^{3}} d x

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{x}{( x - 3 ) ^{3}} d x + \int \frac{1}{( x - 3 ) ^{3}} d x

\int \frac{x}{( x - 3 ) ^{3}} d x = - \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}}

\int \frac{1}{( x - 3 ) ^{3}} d x = - \frac{1}{2 ( x - 3 ) ^{2}}

= - \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}} - \frac{1}{2 ( x - 3 ) ^{2}}

Simplify

- \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}} - \frac{1}{2 ( x - 3 ) ^{2}} : - \frac{1}{x - 3} - \frac{2}{( x - 3 ) ^{2}}

= - \frac{1}{x - 3} - \frac{2}{( x - 3 ) ^{2}}

Add a constant to the solution

= - \frac{1}{x - 3} - \frac{2}{( x - 3 ) ^{2}} + C

\int \frac{x ^{2}}{x ^{2} + 19} d x

\int \frac{x - 2}{x ^{2} + 2 x + 5} d x

\int \frac{3 x ^{2}}{1 - x ^{3}} d x

\int 7 sec^{5} (x) d x

e^{(x^{2})}

What is the integral of (x+1)/((x-3)^3) ?
The integral of (x+1)/((x-3)^3) is -1/(x-3)-2/((x-3)^2)+C