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

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

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

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

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

Solution steps

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

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 = x^{2} + 1

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

Add a constant to the solution

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

y = 8 sin (π x - y), (1, 0)

\frac{d}{d x} ((3 sin (x) + x)^{3})

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

sin (t + \frac{11 π}{2})

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

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