What is the integral of 1/((m+n)+(m-n)x^2) ?

The integral of 1/((m+n)+(m-n)x^2) is 1/(sqrt(m-n)\sqrt{m+n)}arctan(sqrt((m-n)/(m+n))x)+C

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integral of 1/((m+n)+(m-n)x^2)

\int \frac{1}{( m + n ) + ( m - n ) x ^{2}} d x

\frac{1}{m - n m + n} arctan (\frac{m - n}{m + n} x) + C

Solution steps

\int \frac{1}{m + n + ( m - n ) x ^{2}} d x

Apply Integral Substitution

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

Take the constant out:

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

= \frac{1}{m - n m + n} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{m - n m + n} arctan (u)

Substitute back

u = \frac{m - n}{m + n} x

= \frac{1}{m - n m + n} arctan (\frac{m - n}{m + n} x)

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{1}{m - n m + n} arctan (\frac{m - n}{m + n} x)

Add a constant to the solution

= \frac{1}{m - n m + n} arctan (\frac{m - n}{m + n} x) + C

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What is the integral of 1/((m+n)+(m-n)x^2) ?
The integral of 1/((m+n)+(m-n)x^2) is 1/(sqrt(m-n)\sqrt{m+n)}arctan(sqrt((m-n)/(m+n))x)+C