What is the integral of sin(5x)cos(x) ?

The integral of sin(5x)cos(x) is 1/2 (-1/6 cos(6x)-1/4 cos(4x))+C

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integral of sin(5x)cos(x)

\int sin (5 x) cos (x) d x

\frac{1}{2} (- \frac{1}{6} cos (6 x) - \frac{1}{4} cos (4 x)) + C

Solution steps

\int sin (5 x) cos (x) d x

Use the following identity:

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

= \int \frac{sin ( 5 x + x ) + sin ( 5 x - 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 (5 x + x) + sin (5 x - 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 (5 x + x) d x + \int sin (5 x - x) d x)

\int sin (5 x + x) d x = - \frac{1}{6} cos (6 x)

\int sin (5 x - x) d x = - \frac{1}{4} cos (4 x)

= \frac{1}{2} (- \frac{1}{6} cos (6 x) - \frac{1}{4} cos (4 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{6} cos (6 x) - \frac{1}{4} cos (4 x)) + C

\int \frac{x ^{3} - x ^{2} + 2 x - 6}{x} d x

4 x + 7, 2 x + 29, 6, 16

x \to - 4 lim ((1 - x)^{3})

n = 1 \sum \infty (x + 5)^{n}

y = (x^{4} + 1)^{4}

What is the integral of sin(5x)cos(x) ?
The integral of sin(5x)cos(x) is 1/2 (-1/6 cos(6x)-1/4 cos(4x))+C