integral of (a+bsin^3(t))/(cos^2(t))

解答

\int \frac{a + b sin ^{3} ( t )}{cos ^{2} ( t )} d t

a tan (t) + b (sec (t) + cos (t)) + C

求解步骤

\int \frac{a + b sin ^{3} ( t )}{cos ^{2} ( t )} d t

乘开

\frac{a + sin ^{3} ( t ) b}{cos ^{2} ( t )} : \frac{a}{cos ^{2} ( t )} + \frac{b sin ^{3} ( t )}{cos ^{2} ( t )}

= \int \frac{a}{cos ^{2} ( t )} + \frac{b sin ^{3} ( t )}{cos ^{2} ( t )} d t

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{a}{cos ^{2} ( t )} d t + \int \frac{b sin ^{3} ( t )}{cos ^{2} ( t )} d t

\int \frac{a}{cos ^{2} ( t )} d t = a tan (t)

\int \frac{b sin ^{3} ( t )}{cos ^{2} ( t )} d t = b (sec (t) + cos (t))

= a tan (t) + b (sec (t) + cos (t))

解答补常数

= a tan (t) + b (sec (t) + cos (t)) + C

\int - 32 x d x

\int_{0}^{2} \frac{1}{x ^{2} - 6 x + 5} d x

\frac{d}{d x} (arctan (x) - \frac{π}{4})

t an g e n t f (x) = 3 sin (x)

in v erse l a pl a ce \frac{10}{2 s + 1}