integral of cos(4x)cos(x)

解答

\int cos (4 x) cos (x) d x

\frac{1}{2} (\frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) + C

求解步骤

\int cos (4 x) cos (x) d x

使用三角恒等式改写

= \int \frac{1}{2} (cos (5 x) + cos (3 x)) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int cos (5 x) + cos (3 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

= \frac{1}{2} (\int cos (5 x) d x + \int cos (3 x) d x)

\int cos (5 x) d x = \frac{1}{5} sin (5 x)

\int cos (3 x) d x = \frac{1}{3} sin (3 x)

= \frac{1}{2} (\frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x))

解答补常数

= \frac{1}{2} (\frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) + C

\int (x - \frac{1}{x}) d x

\int \frac{y ^{2} + 5}{y ^{2} + 9} d y

\int \frac{x ^{2} + 1}{x ^{3} - x} d x

\int \frac{1}{5 1 - 2 x} d x

\int \frac{- x ^{2}}{4} d x