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

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

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

\int \frac{y ^{3}}{y ^{2} + 1} d y

\frac{1}{2} (y^{2} + 1 - ln y^{2} + 1) + C

Solution steps

\int \frac{y ^{3}}{y ^{2} + 1} d y

Apply u-substitution

: \int \frac{u - 1}{2 u} d u

= \int \frac{u - 1}{2 u} d u

Take the constant out:

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

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

Expand

\frac{u - 1}{u} : 1 - \frac{1}{u}

= \frac{1}{2} \cdot \int 1 - \frac{1}{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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{2} (u - ln ∣ u ∣)

Substitute back

u = y^{2} + 1

= \frac{1}{2} (y^{2} + 1 - ln y^{2} + 1)

Add a constant to the solution

= \frac{1}{2} (y^{2} + 1 - ln y^{2} + 1) + C

\frac{d}{d x} (a x cos (3 x) + b x sin (3 x))

\int \frac{2}{t ^{2}} d t

\int (x^{3} - sin (x)) d x

\int \frac{1}{cos ^{2} ( 5 x )} d x

x \to 1 lim (\frac{2 \cdot x + 3}{x + 4})

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