What is the integral of 3/((x^2+1)^2) ?

The integral of 3/((x^2+1)^2) is 3/2 (arctan(x)+x/(1+x^2))+C

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integral of 3/((x^2+1)^2)

\int \frac{3}{( x ^{2} + 1 ) ^{2}} d x

\frac{3}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

Solution steps

\int \frac{3}{( x ^{2} + 1 ) ^{2}} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{( x ^{2} + 1 ) ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{1}{sec ^{4} ( u ) cos ^{2} ( u )} d u

= 3 \cdot \int \frac{1}{sec ^{4} ( u ) cos ^{2} ( u )} d u

Rewrite using trig identities

= 3 \cdot \int \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= 3 \cdot \frac{1}{2} \cdot \int 1 + cos (2 u) d u

Apply the Sum Rule:

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

= 3 \cdot \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= 3 \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arctan (x)

= 3 \cdot \frac{1}{2} (arctan (x) + \frac{1}{2} sin (2 arctan (x)))

Simplify

3 \cdot \frac{1}{2} (arctan (x) + \frac{1}{2} sin (2 arctan (x))) : \frac{3}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

= \frac{3}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

Add a constant to the solution

= \frac{3}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

\int \frac{in x}{x ^{2}} d x

\frac{d}{d t} (3 sin^{3} (t))

7 \frac{d y}{d x} = y^{2}

f (x) = 3 x^{3} - x^{2} + 4, (1, 6)

\frac{d}{d x} (- \frac{6}{5} e^{- 20 x})

What is the integral of 3/((x^2+1)^2) ?
The integral of 3/((x^2+1)^2) is 3/2 (arctan(x)+x/(1+x^2))+C