integral of sec(x)(5sec(x)+8tan(x))

解答

\int sec (x) (5 sec (x) + 8 tan (x)) d x

5 tan (x) + 8 sec (x) + C

求解步骤

\int sec (x) (5 sec (x) + 8 tan (x)) d x

使用三角恒等式改写

= \int \frac{5 + 8 sin ( x )}{cos ^{2} ( x )} d x

乘开

\frac{5 + 8 sin ( x )}{cos ^{2} ( x )} : \frac{5}{cos ^{2} ( x )} + \frac{8 sin ( x )}{cos ^{2} ( x )}

= \int \frac{5}{cos ^{2} ( x )} + \frac{8 sin ( 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{5}{cos ^{2} ( x )} d x + \int \frac{8 sin ( x )}{cos ^{2} ( x )} d x

\int \frac{5}{cos ^{2} ( x )} d x = 5 tan (x)

\int \frac{8 sin ( x )}{cos ^{2} ( x )} d x = 8 sec (x)

= 5 tan (x) + 8 sec (x)

解答补常数

= 5 tan (x) + 8 sec (x) + C

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