What is the integral of sin(20x)sin(21x) ?

The integral of sin(20x)sin(21x) is 1/2 (sin(x)-1/41 sin(41x))+C

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integral of sin(20x)sin(21x)

\int sin (20 x) sin (21 x) d x

\frac{1}{2} (sin (x) - \frac{1}{41} sin (41 x)) + C

Solution steps

\int sin (20 x) sin (21 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (41 x) + cos (- x)) 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 - cos (41 x) + cos (- x) d x

Simplify

- cos (41 x) + cos (- x) : cos (x) - cos (41 x)

= \frac{1}{2} \cdot \int cos (x) - cos (41 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 cos (x) d x - \int cos (41 x) d x)

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

\int cos (41 x) d x = \frac{1}{41} sin (41 x)

= \frac{1}{2} (sin (x) - \frac{1}{41} sin (41 x))

Add a constant to the solution

= \frac{1}{2} (sin (x) - \frac{1}{41} sin (41 x)) + C

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

\frac{d}{d x} (\frac{4 x ^{2} + 5}{x ^{2} + 7})

\frac{\partial}{\partial y} (10 - 6 y + e^{- 3 x})

\frac{d}{d t} (3 e^{t})

F (x) = \frac{x ^{2}}{4 + x}

What is the integral of sin(20x)sin(21x) ?
The integral of sin(20x)sin(21x) is 1/2 (sin(x)-1/41 sin(41x))+C