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

解答

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

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

求解步骤

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

使用积分加法定则:

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

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

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

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

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

解答补常数

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

\int \frac{1}{x ( 9 x + 4 )} d x

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