What is the integral of x/((5-4x-x^2)^{3/2)} ?

The integral of x/((5-4x-x^2)^{3/2)} is 1/((-x^2-4x+5)^{1/2)}-(2(x+2))/(9(-x^2-4x+5)^{1/2)}+C

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integral of x/((5-4x-x^2)^{3/2)}

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

\frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} + C

Solution steps

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

Complete the square

5 - 4 x - x^{2} : - (x + 2)^{2} + 9

= \int \frac{x}{( - ( x + 2 ) ^{2} + 9 ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{u - 2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u

= \int \frac{u - 2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u

Expand

\frac{u - 2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} : \frac{u}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} - \frac{2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}}

= \int \frac{u}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} - \frac{2}{( - u ^{2} + 9 ) ^{\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} + 9 ) ^{\frac{3}{2}}} d u - \int \frac{2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u

\int \frac{u}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u = \frac{1}{( - u ^{2} + 9 ) ^{\frac{1}{2}}}

\int \frac{2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u = \frac{2 u}{9 ( 9 - u ^{2} ) ^{\frac{1}{2}}}

= \frac{1}{( - u ^{2} + 9 ) ^{\frac{1}{2}}} - \frac{2 u}{9 ( 9 - u ^{2} ) ^{\frac{1}{2}}}

Substitute back

u = x + 2

= \frac{1}{( - ( x + 2 ) ^{2} + 9 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( 9 - ( x + 2 ) ^{2} ) ^{\frac{1}{2}}}

Simplify

\frac{1}{( - ( x + 2 ) ^{2} + 9 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( 9 - ( x + 2 ) ^{2} ) ^{\frac{1}{2}}} : \frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}}

= \frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} + C

\int 6 x - 2 d x

g (w) = \frac{8}{3 w ^{5}} + 9 w

\int sin (\frac{1}{4}) x d x

\int 3 x + 7 d x

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

What is the integral of x/((5-4x-x^2)^{3/2)} ?
The integral of x/((5-4x-x^2)^{3/2)} is 1/((-x^2-4x+5)^{1/2)}-(2(x+2))/(9(-x^2-4x+5)^{1/2)}+C