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

解答

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

tan (x) - ln (cos^{2} (x)) + C

求解步骤

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

乘开

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

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

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

\int \frac{sin ( 2 x )}{cos ^{2} ( x )} d x = - ln (cos^{2} (x))

= tan (x) - ln (cos^{2} (x))

解答补常数

= tan (x) - ln (cos^{2} (x)) + C

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

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