Question

integrate it.

Important for 12th board

​

Answer 1

To integrate :

$\star \ \int \: {sin}^{4} x \: dx$

$\implies \: \int \: ( {sin}^{2} x) ^{2} dx \\ \\ \implies \: \int( \frac{1 - cos2x}{2} ) ^{2} dx \\ \\ \implies \: \frac{1}{4} \int \: (1 - 2cos2x + {cos}^{2} 2x)dx \\ \\ \implies \: \frac{1}{4} \int \: ( \frac{3}{2} - 2cos2x + \frac{1}{2} cos4x) \: dx \\ \\ \implies \: \frac{1}{4} ( \frac{3}{2} x - \frac{2}{2} sin2x + \frac{1}{2} \frac{sin4x}{4} ) + c \\ \\ \implies \: \frac{3}{8} x - \frac{sin2x}{4} + \frac{ sin4x}{32} + c$

Formula used :

$\star \bold{\int \: {x}^{n \: dx \: = \frac{ {x}^{n + 1} }{n + 1} } + c } \\ \\ \star \bold{ \int \: sinx \: dx = - cosx + c} \\ \\ \star \bold{ \int \: cosxdx = sinx \: + c} \\ \\ \star \: \bold{{sin}^{2} x = \frac{1 - cos2x}{2} } \\ \\ \star \: \bold{{cos}^{2} 2x = \frac{cos4x + 1}{2} }$

Answer 2

[Tex]\mathbb{Answer}[/tex]