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

解

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

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

解答ステップ

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

拡張

\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

総和規則を適用する:

\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))

定数を解答に追加する

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