integral of sec(x)(7sec(x)+6tan(x))

解答

\int sec (x) (7 sec (x) + 6 tan (x)) d x

7 tan (x) + 6 sec (x) + C

求解步骤

\int sec (x) (7 sec (x) + 6 tan (x)) d x

使用三角恒等式改写

= \int \frac{7 + 6 sin ( x )}{cos ^{2} ( x )} d x

乘开

\frac{7 + 6 sin ( x )}{cos ^{2} ( x )} : \frac{7}{cos ^{2} ( x )} + \frac{6 sin ( x )}{cos ^{2} ( x )}

= \int \frac{7}{cos ^{2} ( x )} + \frac{6 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{7}{cos ^{2} ( x )} d x + \int \frac{6 sin ( x )}{cos ^{2} ( x )} d x

\int \frac{7}{cos ^{2} ( x )} d x = 7 tan (x)

\int \frac{6 sin ( x )}{cos ^{2} ( x )} d x = 6 sec (x)

= 7 tan (x) + 6 sec (x)

解答补常数

= 7 tan (x) + 6 sec (x) + C

\int 7 x cos (x^{2} - 3) d x

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

\int x (2 x + 3)^{7} d x

\int \frac{cot ( t ) csc ( t )}{csc ^{7} ( t )} d t

\int \frac{2 x - 7}{x ^{2} - 12 x + 32} d x