Question

Differentiate it . important for 12th board .

Find dy/dx if y = sin³x cos^5x​

Answer 1

Answer is given in the pic.

Answer 2

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

$\implies \: { \sin }^{3} x = \sin(x) (1 - { \cos }^{2} x)$

Solution :

$\int \: { \sin }^{3} x \: { \cos }^{5}x \: \\ = \int \sin(x) (1 - { \cos }^{2} x) { \cos}^{5} x \: dx$

Let, u = cos x

$\implies \: du \: = - \sin(x) dx$

$\implies \: sinx \: dx = - du$

$\implies \int( {u}^{5} - {u}^{7}) du$

$\implies \: - ( \dfrac{ {u}^{6} }{6} - \dfrac{ {u}^{8} }{8} ) + c$

$\implies \: \dfrac{ { \cos }^{8}x }{8} - \dfrac{ { \cos}^{6}x}{6} + c$

$\bigstar \: \dfrac{ { \cos }^{8}x }{8} - \dfrac{ { \cos}^{6}x}{6} + c$