What is the integral of 1/((5-4x-x^2)^{5/2)} ?

The integral of 1/((5-4x-x^2)^{5/2)} is (3(x+2)(-x^2-4x+5)+(x+2)^3)/(243(-x^2-4x+5)^{3/2)}+C

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integral of 1/((5-4x-x^2)^{5/2)}

\int \frac{1}{( 5 - 4 x - x ^{2} ) ^{\frac{5}{2}}} d x

\frac{3 ( x + 2 ) ( - x ^{2} - 4 x + 5 ) + ( x + 2 ) ^{3}}{243 ( - x ^{2} - 4 x + 5 ) ^{\frac{3}{2}}} + C

Solution steps

\int \frac{1}{( 5 - 4 x - x ^{2} ) ^{\frac{5}{2}}} d x

Complete the square

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

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

Apply u-substitution

: \int \frac{1}{( - u ^{2} + 9 ) ^{\frac{5}{2}}} d u

= \int \frac{1}{( - u ^{2} + 9 ) ^{\frac{5}{2}}} d u

Apply Trigonometric Substitution

: \int \frac{1}{81 cos ^{4} ( v )} d v

= \int \frac{1}{81 cos ^{4} ( v )} d v

Take the constant out:

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

= \frac{1}{81} \cdot \int \frac{1}{cos ^{4} ( v )} d v

Rewrite using trig identities

= \frac{1}{81} \cdot \int (1 + tan^{2} (v)) sec^{2} (v) d v

Apply u-substitution

: \int 1 + w^{2} d w

= \frac{1}{81} \cdot \int 1 + w^{2} d w

Apply the Sum Rule:

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

= \frac{1}{81} (\int 1 d w + \int w^{2} d w)

\int 1 d w = w

\int w^{2} d w = \frac{w ^{3}}{3}

= \frac{1}{81} (w + \frac{w ^{3}}{3})

Substitute back

= \frac{1}{81} (tan (arcsin (\frac{1}{3} (x + 2))) + \frac{tan ^{3} ( arcsin ( \frac{1}{3} ( x + 2 ) ) )}{3})

Simplify

\frac{1}{81} (tan (arcsin (\frac{1}{3} (x + 2))) + \frac{tan ^{3} ( arcsin ( \frac{1}{3} ( x + 2 ) ) )}{3}) : \frac{3 ( x + 2 ) ( - x ^{2} - 4 x + 5 ) + ( x + 2 ) ^{3}}{243 ( - x ^{2} - 4 x + 5 ) ^{\frac{3}{2}}}

= \frac{3 ( x + 2 ) ( - x ^{2} - 4 x + 5 ) + ( x + 2 ) ^{3}}{243 ( - x ^{2} - 4 x + 5 ) ^{\frac{3}{2}}}

Add a constant to the solution

= \frac{3 ( x + 2 ) ( - x ^{2} - 4 x + 5 ) + ( x + 2 ) ^{3}}{243 ( - x ^{2} - 4 x + 5 ) ^{\frac{3}{2}}} + C

\int \frac{1}{cos ( x ) + sin ( x )} d x

\frac{5}{( s + 2 ) ^{4}}

\frac{d}{d x} (a (x + 1) (x - r (x)))

\int \frac{- 2 \cdot x ^{4}}{2 \cdot x ^{3} - 3 \cdot x ^{2}} d x

\int \frac{x ^{4} + x - 25}{x ^{2} + 5} d x

What is the integral of 1/((5-4x-x^2)^{5/2)} ?
The integral of 1/((5-4x-x^2)^{5/2)} is (3(x+2)(-x^2-4x+5)+(x+2)^3)/(243(-x^2-4x+5)^{3/2)}+C