Question

$\large \displaystyle \sf \red{\int \frac{ \sin ^{8}x - {cos}^{8} x}{1 - {2sin}^{2} x \: {cos}^{2}x } }$​

Answer 1

I HOPE THIS WILL BE THE BEST ANSWER FOR YOU ❤️❤️

Answer 2

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:\displaystyle\int\rm \frac{ {sin}^{8}x - {cos}^{8} x}{1 - 2 {sin}^{2}x \: {cos}^{2} x} \: dx$

can be rewritten as

$\rm \: = \: \displaystyle\int\rm \frac{ {( {sin}^{2} x)}^{4} - {( {cos}^{2}x)}^{4} }{1 - 2 {sin}^{2}x \: {cos}^{2}x} \: x$

We know, from algebraic Identities

$\red{\boxed{\tt{ \: {x}^{4} - {y}^{4} = (x - y)(x + y)( {x}^{2} + {y}^{2}) \: }}}$

So, using this identity, we get

$\rm \: = \: \displaystyle\int\rm \frac{( {sin}^{2}x - {cos}^{2}x)( {sin}^{2}x + {cos}^{2}x)( {sin}^{4}x + {cos}^{4}x)}{1 - 2 {sin}^{2}x \: {cos}^{2}x} \: dx$

We know,

$\boxed{\tt{ {sin}^{2}x + {cos}^{2}x = 1}}$

So, we get

$\rm \: = \: - \: \displaystyle\int\rm \frac{( {cos}^{2}x - {sin}^{2}x)(1)[ {( {sin}^{2}x) }^{2} + {( {cos}^{2}x)}^{2}]}{1 - 2 {sin}^{2}x \: {cos}^{2} x} \: dx$

We know,

$\boxed{\tt{ {x}^{2} + {y}^{2} = {(x + y)}^{2} - 2xy \: }}$

and

$\boxed{\tt{ {cos}^{2}x - {sin}^{2}x = cos2x}}$

So, using this, we get

$\rm \: = \: - \: \displaystyle\int\rm \frac{cos2x[ {( {cos}^{2}x + {sin}^{2}x)}^{2} - 2 {sin}^{2} x \: {cos}^{2}x] }{1 - 2 {sin}^{2}x \: {cos}^{2} x} \: dx$

$\rm \: = \: - \: \displaystyle\int\rm \frac{cos2x[ {( 1)}^{2} - 2 {sin}^{2} x \: {cos}^{2}x] }{1 - 2 {sin}^{2}x \: {cos}^{2} x} \: dx$

$\rm \: = \: - \: \displaystyle\int\rm \frac{cos2x[ 1 - 2 {sin}^{2} x \: {cos}^{2}x] }{1 - 2 {sin}^{2}x \: {cos}^{2} x} \: dx$

$\rm \: = \: - \: \displaystyle\int\rm cos2x \: dx$

$\rm \: = \: - \: \dfrac{sin2x}{2} + c$

Hence,

$\rm :\longmapsto\:\boxed{\tt{ \: \displaystyle\int\rm \frac{ {sin}^{8}x - {cos}^{8} x}{1 - 2 {sin}^{2}x \: {cos}^{2} x} \: dx = - \frac{sin2x}{2} + c \: }}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

EXPLORE MORE

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$