What is the integral of 1/(9-x^2) ?

The integral of 1/(9-x^2) is 1/6 (ln| x/3+1|-ln| x/3-1|)+C

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integral of 1/(9-x^2)

\int \frac{1}{9 - x ^{2}} d x

\frac{1}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1) + C

Solution steps

\int \frac{1}{9 - x ^{2}} d x

Apply Integral Substitution

= \int \frac{1}{3 ( - 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}{3} \cdot \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}{3} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Substitute back

u = \frac{x}{3}

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

Simplify

\frac{1}{3} (\frac{ln \frac{x}{3} + 1}{2} - \frac{ln \frac{x}{3} - 1}{2}) : \frac{1}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)

= \frac{1}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)

Add a constant to the solution

= \frac{1}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1) + C

\int e^{- k N x} d x

y = A x^{2} + B e^{2 x}

y^{^{'}} = 2 x y, y (0) = 1

(x y) d x + (x^{2} + y^{2}) d y = 0

What is the integral of 1/(9-x^2) ?
The integral of 1/(9-x^2) is 1/6 (ln| x/3+1|-ln| x/3-1|)+C