Question

Please solve this ASAP ​

Answer 1

The Correct and Right Answer is in The Above Attachment....

Hope It Helps...

Answer 2

Question

$\sf = > \displaystyle \int \dfrac{ {sin}^{8}x - {cos}^{8} x}{1 - 2 {sin}^{2} x {cos}^{2} x} dx$

$\huge\bold{\gray{\sf{Answer:}}}$

$\bold{Explanation:}$

Identity used here :-

$\sf\boxed{{x}^{2} - {y}^{2} = (x - y)(x + y)}$

$\sf= > {sin}^{8} x - {cos}^{8} x = { {(sin}^{4}x) }^{2} - { {(cos}^{4}x) }^{2}$

$\sf= >( {sin}^{4} x - {cos}^{4}x)( {sin}^{4} x + {cos}^{4} x)$

$\sf= > ( {sin}^{2} x - {cos}^{2} x)( {sin}^{2} x + {cos}^{2} x)$

$\bigg[ { {\bigg(sin}^{2}x\bigg) }^{2} + {\bigg( {cos}^{2}x\bigg )}^{2} + 2 {cos}^{4} x {sin}^{2} x - 2 {cos}^{2} x {sin}^{2} x\bigg]$

$\sf= > {sin}^{2} x - {cos}^{2} x(1)\bigg[ {( {sin}^{2}x + {cos}^{2}x) }^{2} - 2 {cos}^{2} x {sin}^{2} x\bigg]$

$\sf = {sin}^{2} x - {cos}^{2} x(1 - 2 {cos}^{2} x {sin}^{2} x)$

$\sf{sin}^{8} x - {cos}^{8} x = {sin}^{2} x - {cos}^{2} x(1 - 2 {cos}^{2}x {sin}^{2} x)$

Dividing both sides by 1-2sin²xcos²x

$\sf= > - \bigg( {cos}^{2} x - {sin}^{2} x\bigg)$

$\sf= > \displaystyle \int - {cos}^{2} xdx$

$\sf = > \displaystyle- \int \: cos2xdx$

$\sf\bold{\red{ = - \dfrac{sin2x}{2} + c}}$