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

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

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

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

\frac{1}{( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}} + \frac{3 ( x - 1 )}{4 ( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}} + C

Solution steps

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

Complete the square

3 + 2 x - x^{2} : - (x - 1)^{2} + 4

= \int \frac{x + 2}{( - ( x - 1 ) ^{2} + 4 ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{u + 3}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} d u

= \int \frac{u + 3}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} d u

Expand

\frac{u + 3}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} : \frac{u}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} + \frac{3}{( - u ^{2} + 4 ) ^{\frac{3}{2}}}

= \int \frac{u}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} + \frac{3}{( - u ^{2} + 4 ) ^{\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} + 4 ) ^{\frac{3}{2}}} d u + \int \frac{3}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} d u

\int \frac{u}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} d u = \frac{1}{( - u ^{2} + 4 ) ^{\frac{1}{2}}}

\int \frac{3}{( - u ^{2} + 4 ) ^{\frac{3}{2}}} d u = \frac{3 u}{4 ( 4 - u ^{2} ) ^{\frac{1}{2}}}

= \frac{1}{( - u ^{2} + 4 ) ^{\frac{1}{2}}} + \frac{3 u}{4 ( 4 - u ^{2} ) ^{\frac{1}{2}}}

Substitute back

u = x - 1

= \frac{1}{( - ( x - 1 ) ^{2} + 4 ) ^{\frac{1}{2}}} + \frac{3 ( x - 1 )}{4 ( 4 - ( x - 1 ) ^{2} ) ^{\frac{1}{2}}}

Simplify

\frac{1}{( - ( x - 1 ) ^{2} + 4 ) ^{\frac{1}{2}}} + \frac{3 ( x - 1 )}{4 ( 4 - ( x - 1 ) ^{2} ) ^{\frac{1}{2}}} : \frac{1}{( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}} + \frac{3 ( x - 1 )}{4 ( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}}

= \frac{1}{( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}} + \frac{3 ( x - 1 )}{4 ( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{1}{( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}} + \frac{3 ( x - 1 )}{4 ( - x ^{2} + 2 x + 3 ) ^{\frac{1}{2}}} + C

\frac{d}{d x} (a e^{b x})

x \to 2 lim (2 x)

\int \frac{100}{7 x - 16} d x

\frac{d}{d x} (\frac{4}{x + 2 x})

f (x) = 3 x^{\frac{2}{3}}

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