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

The integral of (x')/(x^2+1) is (xx')/(x^2+1)-1/(x^2+1)+C

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

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

\frac{xx '}{x ^{2} + 1} - \frac{1}{x ^{2} + 1} + C

Solution steps

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

Apply Integration By Parts

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

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

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

= \frac{x x ^{^{'}}}{x ^{2} + 1} - \frac{1}{x ^{2} + 1}

Add a constant to the solution

= \frac{x x ^{^{'}}}{x ^{2} + 1} - \frac{1}{x ^{2} + 1} + C

\int x^{17} sin (x^{18}) d x

g (t) = \frac{1 - 3 t}{4 + t}

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

\int (e^{- \frac{x}{4}} + e^{\frac{x}{2}}) d x

(x + 1) \frac{d y}{d x} = (x + 6)

What is the integral of (x')/(x^2+1) ?
The integral of (x')/(x^2+1) is (xx')/(x^2+1)-1/(x^2+1)+C