Question

find the solution for the image​

Answer 1

$\huge\mathfrak{solution}$

To find:

$\int \: \frac{ \sin 3x}{ \sin x} dx$

As we know that :

$\star \boxed{ \bold{\sin 3x = 3 \sin x - 4 { \sin}^{3} x}}$

$\implies \int \: \frac{ \sin 3x}{ \sin x} dx \\ \\ \implies \int \frac{3 \sin x - 4 { \sin}^{3}x }{ \sin x} dx \\ \\ \implies \int \: \frac{3 \sin x}{ \sin x} dx - \int \: \frac{4 { \sin}^{3}x }{ \sin x} dx \\ \\ \implies \int \: 3dx \: - \int \: 4 { \sin}^{2} x \: dx \\ \\ \implies \: 3\int \: dx - 4 \int \: { \sin }^{2} dx$

$\star \boxed{ \bold{ { \sin}^{2} x = \frac{1 - \cos \: 2x}{2}}}$

$\implies \: 3 \int \: dx - 4 \int \: \frac{1 - \cos 2x}{2} dx \\ \\ \implies \: 3\int dx - 2 \int \: (1 - \cos \: 2x )\: dx \\ \\ \implies3 \int \: dx - 2 \int \: dx + 2 \int \cos 2x \: dx \\ \\ \implies \: 3x - 2x + 2 \frac{ \sin 2x}{2} \\ \\ \implies \: x + \sin 2x \: + c$

Formula:

$\star \bold{ \int \cos \: ax \: dx = \frac{ \sin ax}{a} + c}$

Answer 2

Step-by-step explanation:

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