What is the integral of (x-1)/((-x^2-2x+15)^{3/2)} ?

The integral of (x-1)/((-x^2-2x+15)^{3/2)} is 1/((-x^2-2x+15)^{1/2)}-(x+1)/(8(-x^2-2x+15)^{1/2)}+C

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integral of (x-1)/((-x^2-2x+15)^{3/2)}

\int \frac{x - 1}{( - x ^{2} - 2 x + 15 ) ^{\frac{3}{2}}} d x

\frac{1}{( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}} - \frac{x + 1}{8 ( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{x - 1}{( - x ^{2} - 2 x + 15 ) ^{\frac{3}{2}}} d x

Complete the square

- x^{2} - 2 x + 15 : - (x + 1)^{2} + 16

= \int \frac{x - 1}{( - ( x + 1 ) ^{2} + 16 ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{u - 2}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} d u

= \int \frac{u - 2}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} d u

Expand

\frac{u - 2}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} : \frac{u}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} - \frac{2}{( - u ^{2} + 16 ) ^{\frac{3}{2}}}

= \int \frac{u}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} - \frac{2}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} 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{u}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} d u - \int \frac{2}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} d u

\int \frac{u}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} d u = \frac{1}{( - u ^{2} + 16 ) ^{\frac{1}{2}}}

\int \frac{2}{( - u ^{2} + 16 ) ^{\frac{3}{2}}} d u = \frac{u}{8 ( - u ^{2} + 16 ) ^{\frac{1}{2}}}

= \frac{1}{( - u ^{2} + 16 ) ^{\frac{1}{2}}} - \frac{u}{8 ( - u ^{2} + 16 ) ^{\frac{1}{2}}}

Substitute back

u = x + 1

= \frac{1}{( - ( x + 1 ) ^{2} + 16 ) ^{\frac{1}{2}}} - \frac{x + 1}{8 ( - ( x + 1 ) ^{2} + 16 ) ^{\frac{1}{2}}}

Simplify

\frac{1}{( - ( x + 1 ) ^{2} + 16 ) ^{\frac{1}{2}}} - \frac{x + 1}{8 ( - ( x + 1 ) ^{2} + 16 ) ^{\frac{1}{2}}} : \frac{1}{( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}} - \frac{x + 1}{8 ( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}}

= \frac{1}{( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}} - \frac{x + 1}{8 ( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{1}{( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}} - \frac{x + 1}{8 ( - x ^{2} - 2 x + 15 ) ^{\frac{1}{2}}} + C

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What is the integral of (x-1)/((-x^2-2x+15)^{3/2)} ?
The integral of (x-1)/((-x^2-2x+15)^{3/2)} is 1/((-x^2-2x+15)^{1/2)}-(x+1)/(8(-x^2-2x+15)^{1/2)}+C