What is the integral of sin(x/5)sin(x/2) ?

The integral of sin(x/5)sin(x/2) is 1/2 (10/3 sin((3x)/(10))-10/7 sin((7x)/(10)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin(x/5)sin(x/2)

\int sin (\frac{x}{5}) sin (\frac{x}{2}) d x

\frac{1}{2} (\frac{10}{3} sin (\frac{3 x}{10}) - \frac{10}{7} sin (\frac{7 x}{10})) + C

Solution steps

\int sin (\frac{x}{5}) sin (\frac{x}{2}) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (\frac{7 x}{10}) + cos (- \frac{3 x}{10})) 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 (\frac{7 x}{10}) + cos (- \frac{3 x}{10}) d x

Simplify

- cos (\frac{7 x}{10}) + cos (- \frac{3 x}{10}) : cos (\frac{3 x}{10}) - cos (\frac{7 x}{10})

= \frac{1}{2} \cdot \int cos (\frac{3 x}{10}) - cos (\frac{7 x}{10}) 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 (\frac{3 x}{10}) d x - \int cos (\frac{7 x}{10}) d x)

\int cos (\frac{3 x}{10}) d x = \frac{10}{3} sin (\frac{3 x}{10})

\int cos (\frac{7 x}{10}) d x = \frac{10}{7} sin (\frac{7 x}{10})

= \frac{1}{2} (\frac{10}{3} sin (\frac{3 x}{10}) - \frac{10}{7} sin (\frac{7 x}{10}))

Add a constant to the solution

= \frac{1}{2} (\frac{10}{3} sin (\frac{3 x}{10}) - \frac{10}{7} sin (\frac{7 x}{10})) + C

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

y = x^{3} + 4 x, (1, 5)

x \to - 10 lim (∣ x + 10 ∣ + 3)

f (x) = \frac{1}{x ^{5}} - x^{2} - 2 x, at x = 1

f (x) = \frac{x ^{2}}{e ^{x}}

What is the integral of sin(x/5)sin(x/2) ?
The integral of sin(x/5)sin(x/2) is 1/2 (10/3 sin((3x)/(10))-10/7 sin((7x)/(10)))+C