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

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

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

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

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

Solution steps

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

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

assuming

(x^{2} + 1) \geq 0

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

Expand

\frac{x ^{2} - x + 1}{( x ^{2} + 1 ) ^{\frac{3}{2}}} : \frac{x ^{2}}{( x ^{2} + 1 ) ^{\frac{3}{2}}} - \frac{x}{( x ^{2} + 1 ) ^{\frac{3}{2}}} + \frac{1}{( x ^{2} + 1 ) ^{\frac{3}{2}}}

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

Apply the Sum Rule:

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

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

3 e^{x} + \frac{8}{3 x}

\int \frac{6}{1 - x ^{2}} d x

\frac{d}{d x} (ln (π))

n = 1 \sum \infty n^{- \frac{1}{3}}

\int \frac{x ^{3} - x ^{2}}{x ^{2}} d x

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