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

解答

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

2 arctan (x) + \frac{1}{4 e} ((e^{2} - 1) Ci (i - x) + (e^{2} - 1) Ci (x + i) + i (1 + e^{2}) (Si (i - x) + Si (x + i))) + C

求解步骤

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

乘开

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

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

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

\int \frac{sin ( x )}{1 + x ^{2}} d x = \frac{1}{4 e} ((e^{2} - 1) Ci (i - x) + (e^{2} - 1) Ci (x + i) + i (1 + e^{2}) (Si (i - x) + Si (x + i)))

= 2 arctan (x) + \frac{1}{4 e} ((e^{2} - 1) Ci (i - x) + (e^{2} - 1) Ci (x + i) + i (1 + e^{2}) (Si (i - x) + Si (x + i)))

解答补常数

= 2 arctan (x) + \frac{1}{4 e} ((e^{2} - 1) Ci (i - x) + (e^{2} - 1) Ci (x + i) + i (1 + e^{2}) (Si (i - x) + Si (x + i))) + C

\int (cos^{2} (4 x)) d x

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