What is the integral of (-2a^2x-3a)/(a^2x^2-5ax+6) ?

The integral of (-2a^2x-3a)/(a^2x^2-5ax+6) is -ln|4a^2x^2-20ax+24|+8ln|2ax-4|-8ln|2ax-6|+C

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integral of (-2a^2x-3a)/(a^2x^2-5ax+6)

\int \frac{- 2 a ^{2} x - 3 a}{a ^{2} x ^{2} - 5 a x + 6} d x

- ln 4 a^{2} x^{2} - 20 a x + 24 + 8 ln ∣ 2 a x - 4 ∣ - 8 ln ∣ 2 a x - 6 ∣ + C

Solution steps

\int \frac{- 2 a ^{2} x - 3 a}{a ^{2} x ^{2} - 5 a x + 6} d x

Complete the square

a^{2} x^{2} - 5 a x + 6 : (x a^{2} - \frac{5 a}{2 a ^{2}})^{2} + 6 - \frac{25}{4}

= \int \frac{- 2 a ^{2} x - 3 a}{( x a ^{2} - \frac{5 a}{2 a ^{2}} ) ^{2} + 6 - \frac{25}{4}} d x

Apply u-substitution

: \int - \frac{4 ( 2 u + 8 )}{4 u ^{2} - 1} d u

= \int - \frac{4 ( 2 u + 8 )}{4 u ^{2} - 1} d u

Take the constant out:

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

= - 4 \cdot \int \frac{2 u + 8}{4 u ^{2} - 1} d u

Expand

\frac{2 u + 8}{4 u ^{2} - 1} : \frac{2 u}{4 u ^{2} - 1} + \frac{8}{4 u ^{2} - 1}

Apply the Sum Rule:

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

= - 4 (\int \frac{2 u}{4 u ^{2} - 1} d u + \int \frac{8}{4 u ^{2} - 1} d u)

\int \frac{2 u}{4 u ^{2} - 1} d u = \frac{1}{4} ln 4 u^{2} - 1

\int \frac{8}{4 u ^{2} - 1} d u = - 4 (\frac{1}{2} ln ∣ 2 u + 1 ∣ - \frac{1}{2} ln ∣ 2 u - 1 ∣)

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

Substitute back

u = x a^{2} - \frac{5 a}{2 a ^{2}}

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

Simplify

- 4 (\frac{1}{4} ln 4 (x a^{2} - \frac{5 a}{2 a ^{2}})^{2} - 1 - 4 (\frac{1}{2} ln 2 (x a^{2} - \frac{5 a}{2 a ^{2}}) + 1 - \frac{1}{2} ln 2 (x a^{2} - \frac{5 a}{2 a ^{2}}) - 1)) : - ln 4 a^{2} x^{2} - 20 a x + 24 + 8 ln ∣ 2 a x - 4 ∣ - 8 ln ∣ 2 a x - 6 ∣

= - ln 4 a^{2} x^{2} - 20 a x + 24 + 8 ln ∣ 2 a x - 4 ∣ - 8 ln ∣ 2 a x - 6 ∣

Add a constant to the solution

= - ln 4 a^{2} x^{2} - 20 a x + 24 + 8 ln ∣ 2 a x - 4 ∣ - 8 ln ∣ 2 a x - 6 ∣ + C

\int \frac{3}{t ^{3}} d t

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

\int 9 x^{8} e^{7 x^{9}} d x

\int \frac{x ^{5}}{20} d x

What is the integral of (-2a^2x-3a)/(a^2x^2-5ax+6) ?
The integral of (-2a^2x-3a)/(a^2x^2-5ax+6) is -ln|4a^2x^2-20ax+24|+8ln|2ax-4|-8ln|2ax-6|+C