What is the integral of 1/(a^2x^2+8ax+15) ?

The integral of 1/(a^2x^2+8ax+15) is -1/(2a)(ln|ax+5|-ln|ax+3|)+C

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integral of 1/(a^2x^2+8ax+15)

\int \frac{1}{a ^{2} x ^{2} + 8 a x + 15} d x

- \frac{1}{2 a} (ln ∣ a x + 5 ∣ - ln ∣ a x + 3 ∣) + C

Solution steps

\int \frac{1}{a ^{2} x ^{2} + 8 a x + 15} d x

Complete the square

a^{2} x^{2} + 8 a x + 15 : (x a^{2} + \frac{4 a}{a ^{2}})^{2} + 15 - 16

= \int \frac{1}{( x a ^{2} + \frac{4 a}{a ^{2}} ) ^{2} + 15 - 16} d x

Apply u-substitution

: \int \frac{1}{a ( u ^{2} - 1 )} d u

= \int \frac{1}{a ( u ^{2} - 1 )} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{1}{u ^{2} - 1} d u

Factor

u^{2} - 1 : - (- u^{2} + 1)

= \frac{1}{a} \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = x a^{2} + \frac{4 a}{a ^{2}}

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

Simplify

\frac{1}{a} - \frac{ln x a ^{2} + \frac{4 a}{a ^{2}} + 1}{2} - \frac{ln x a ^{2} + \frac{4 a}{a ^{2}} - 1}{2} : - \frac{1}{2 a} (ln ∣ a x + 5 ∣ - ln ∣ a x + 3 ∣)

= - \frac{1}{2 a} (ln ∣ a x + 5 ∣ - ln ∣ a x + 3 ∣)

Add a constant to the solution

= - \frac{1}{2 a} (ln ∣ a x + 5 ∣ - ln ∣ a x + 3 ∣) + C

\int \frac{x - 2}{x + 2} d x

f (x) = (x - 2) (x^{2} - 10 x + 25)

e^{t} sin (t)

f (t) = t^{2} + 1

n = 8 \sum \infty e^{5 - 6 n}

What is the integral of 1/(a^2x^2+8ax+15) ?
The integral of 1/(a^2x^2+8ax+15) is -1/(2a)(ln|ax+5|-ln|ax+3|)+C