Subjective Type

Find the integrals of the functions $$\sin 4x \sin 8x$$

Solution

It is known that, $$\sin A \sin B=\dfrac {1}{2}[\cos (A-B)- \cos (A+B)]$$
$$\therefore\displaystyle \int \sin 4x \sin 8x dx=\int \left \{\frac {1}{2}[\cos (4x-8x)-\cos (4x+8x)]\right \}dx$$
$$\displaystyle =\frac {1}{2}\int (\cos (-4x)-\cos 12x)dx$$
$$\displaystyle =\frac {1}{2}\int (\cos 4x-\cos 12x)dx$$
$$\displaystyle =\frac {1}{2}\left [\frac {\sin 4x}{4}-\frac {\sin 12x}{12}\right ]+C$$

SIMILAR QUESTIONS

Indefinite Integrals

$$\displaystyle \int \frac {\sin^8 x-\cos^8 x}{(1-2\sin^2x \cos^2x)}dx$$ is equal is to

Indefinite Integrals

Let $$\alpha \epsilon (0, \pi/2)$$ be fixed. If the integral $$\int \dfrac {\tan x + \tan \alpha}{\tan x - \tan \alpha} dx =$$ $$A (x) \cos 2\alpha + B(x) \sin 2\alpha + C$$, where $$C$$ is a constant of integration, then the functions $$A(x)$$ and $$B(x)$$ are respectively.

Indefinite Integrals

$$\int \dfrac {dx}{\cos x + \sqrt {3}\sin x}$$ equals

Indefinite Integrals

Find the integral of $$\displaystyle \int \frac {\sec^2x}{co\sec^2x}dx$$

Indefinite Integrals

Find the integral of $$\displaystyle \int \frac {2-3 \sin x}{\cos^2x}dx$$

Indefinite Integrals

Integrate the function $$\sin (ax + b) \cos (ax + b)$$

Indefinite Integrals

Integrate the function $$\displaystyle \frac {2\cos x-3 \sin x}{6 \cos x+4 \sin x}$$

Indefinite Integrals

$$\displaystyle \int \frac {dx}{\sin^2 x \cos^2x}$$ equals

Indefinite Integrals

$$\int_{\cos{2x}\cos{4x}\cos{6x}dx}$$

Indefinite Integrals

Find the integrals of the functions $$\sin x \sin 2x \sin 3x$$

Contact Details