What is the integral of sin(16x)cos(13x) ?

The integral of sin(16x)cos(13x) is 1/2 (-1/29 cos(29x)-1/3 cos(3x))+C

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integral of sin(16x)cos(13x)

\int sin (16 x) cos (13 x) d x

\frac{1}{2} (- \frac{1}{29} cos (29 x) - \frac{1}{3} cos (3 x)) + C

Solution steps

\int sin (16 x) cos (13 x) d x

Use the following identity:

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

= \int \frac{sin ( 16 x + 13 x ) + sin ( 16 x - 13 x )}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (16 x + 13 x) + sin (16 x - 13 x) d x

Simplify

= \frac{1}{2} \cdot \int sin (29 x) + sin (3 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int sin (29 x) d x + \int sin (3 x) d x)

\int sin (29 x) d x = - \frac{1}{29} cos (29 x)

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

= \frac{1}{2} (- \frac{1}{29} cos (29 x) - \frac{1}{3} cos (3 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{29} cos (29 x) - \frac{1}{3} cos (3 x)) + C

\int_{0}^{1} 13 e^{x^{2}} d x

x \to 4 lim (- x + 4)

\int (4 e^{x} + 12 x^{\frac{3}{2}} + 4 x^{\frac{1}{2}}) d x

x^{2} - 4, at x = 3

\frac{ln ( x )}{10 x}

What is the integral of sin(16x)cos(13x) ?
The integral of sin(16x)cos(13x) is 1/2 (-1/29 cos(29x)-1/3 cos(3x))+C