What is the integral of 1/((a+b)+(a-b)x^2) ?

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

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integral of 1/((a+b)+(a-b)x^2)

\int \frac{1}{( a + b ) + ( a - b ) x ^{2}} d x

\frac{1}{a - b a + b} arctan (\frac{a - b}{a + b} x) + C

Solution steps

\int \frac{1}{a + b + ( a - b ) x ^{2}} d x

Apply Integral Substitution

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

Take the constant out:

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

= \frac{1}{a - b a + b} \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}{a - b a + b} arctan (u)

Substitute back

u = \frac{a - b}{a + b} x

= \frac{1}{a - b a + b} arctan (\frac{a - b}{a + b} x)

Combine same powers :

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

= \frac{1}{a - b a + b} arctan (\frac{a - b}{a + b} x)

Add a constant to the solution

= \frac{1}{a - b a + b} arctan (\frac{a - b}{a + b} x) + C

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\frac{d y}{d x} = 4 x^{5} y - y

\int_{- 2}^{0} (\frac{y}{2} + y + 3) d y

e^{- 2 t} sin (5 t)

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