integral of cos(7x)sin(3x)

解答

\int cos (7 x) sin (3 x) d x

\frac{1}{2} (- \frac{1}{10} cos (10 x) + \frac{1}{4} cos (4 x)) + C

求解步骤

\int cos (7 x) sin (3 x) d x

利用以下特性:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int \frac{sin ( 3 x + 7 x ) + sin ( 3 x - 7 x )}{2} 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 sin (3 x + 7 x) + sin (3 x - 7 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 sin (3 x + 7 x) d x + \int sin (3 x - 7 x) d x)

\int sin (3 x + 7 x) d x = - \frac{1}{10} cos (10 x)

\int sin (3 x - 7 x) d x = \frac{1}{4} cos (4 x)

= \frac{1}{2} (- \frac{1}{10} cos (10 x) + \frac{1}{4} cos (4 x))

解答补常数

= \frac{1}{2} (- \frac{1}{10} cos (10 x) + \frac{1}{4} cos (4 x)) + C

\int \frac{ln ( x )}{sec ( c )} d x

\int x 2 x^{2} + 11 d x

\int sin^{2} (x) cos^{2} (d) x d x

\int (x y e^{x z} + z^{3}) d z

\int cos (2 t) d t