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

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

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

\int \frac{π x - 1}{x ^{2} + π ^{2}} d x

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

Solution steps

\int \frac{π x - 1}{x ^{2} + π ^{2}} d x

Expand

\frac{π x - 1}{x ^{2} + π ^{2}} : \frac{π x}{x ^{2} + π ^{2}} - \frac{1}{x ^{2} + π ^{2}}

= \int \frac{π x}{x ^{2} + π ^{2}} - \frac{1}{x ^{2} + π ^{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{π x}{x ^{2} + π ^{2}} d x - \int \frac{1}{x ^{2} + π ^{2}} d x

\int \frac{π x}{x ^{2} + π ^{2}} d x = π x^{2} + π^{2}

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

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

Add a constant to the solution

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

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

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

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

\int \frac{x ^{2} - 4}{3 x ^{3}} d x

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