What is the integral of (x^3)/(sqrt(x^4+1)) ?

The integral of (x^3)/(sqrt(x^4+1)) is 1/2 sqrt(x^4+1)+C

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integral of (x^3)/(sqrt(x^4+1))

\int \frac{x ^{3}}{x ^{4} + 1} d x

\frac{1}{2} x^{4} + 1 + C

Solution steps

\int \frac{x ^{3}}{x ^{4} + 1} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{4} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = x^{4} + 1

= \frac{1}{4} \cdot 2 (x^{4} + 1)^{\frac{1}{2}}

Simplify

\frac{1}{4} \cdot 2 (x^{4} + 1)^{\frac{1}{2}} : \frac{1}{2} x^{4} + 1

= \frac{1}{2} x^{4} + 1

Add a constant to the solution

= \frac{1}{2} x^{4} + 1 + C

x \to 0 lim (\frac{tan ( x )}{5 x})

y^{^{'''}} - 6 y^{^{''}} + 12 y^{^{'}} - 8 y = 0

(e^{x} cos (y) + 2 x) d x - (e^{x} sin (y)) d y = 0

x^{^{'}} = (x - 4)^{2}

\frac{( e ^{2 p} + 1 )}{3 e ^{2 p}}

What is the integral of (x^3)/(sqrt(x^4+1)) ?
The integral of (x^3)/(sqrt(x^4+1)) is 1/2 sqrt(x^4+1)+C