What is the integral of sin(2pix)cos(5pix) ?

The integral of sin(2pix)cos(5pix) is 1/2 (-1/(7pi)cos(7pix)+1/(3pi)cos(3pix))+C

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integral of sin(2pix)cos(5pix)

\int sin (2 π x) cos (5 π x) d x

\frac{1}{2} (- \frac{1}{7 π} cos (7 π x) + \frac{1}{3 π} cos (3 π x)) + C

Solution steps

\int sin (2 π x) cos (5 π x) d x

Use the following identity:

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

= \int \frac{sin ( 2 π x + 5 π x ) + sin ( 2 π x - 5 π 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 (2 π x + 5 π x) + sin (2 π x - 5 π x) d x

Simplify

sin (2 π x + 5 π x) + sin (2 π x - 5 π x) : sin (7 π x) - sin (3 π x)

= \frac{1}{2} \cdot \int sin (7 π x) - sin (3 π 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 (7 π x) d x - \int sin (3 π x) d x)

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

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

= \frac{1}{2} (- \frac{1}{7 π} cos (7 π x) - (- \frac{1}{3 π} cos (3 π x)))

Simplify

= \frac{1}{2} (- \frac{1}{7 π} cos (7 π x) + \frac{1}{3 π} cos (3 π x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{7 π} cos (7 π x) + \frac{1}{3 π} cos (3 π x)) + C

\frac{d}{d x} (ln (\frac{x - 3}{x - 1}))

x \cdot \frac{d y}{d x} = 1 - y^{2}

\int tan (θ) d θ

\int x (x - 1)^{10} d x

(x^{3} \cdot cos (x))^{^{'}}

What is the integral of sin(2pix)cos(5pix) ?
The integral of sin(2pix)cos(5pix) is 1/2 (-1/(7pi)cos(7pix)+1/(3pi)cos(3pix))+C