Question

integral sec ^3/7 co sec ^
11/7 do:​

Answer 1

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

$\displaystyle\rm :\longmapsto\: \int {sec}^{ \frac{3}{7}}x \: {cosec}^{ \frac{11}{7}}x \: dx$

$\displaystyle\rm \: \: = \: \int\dfrac{dx}{ {cos}^{ \frac{3}{7}}x \: {sin}^{ \frac{11}{7}}x}$

Since,

$\rm :\longmapsto\:\dfrac{3}{7} + \dfrac{11}{7} = \dfrac{3 + 11}{7} = \dfrac{14}{7} = 2 \: (even)$

So,

$\red{ \tt \: Divide \: numerator \: and \: denominator \: by \: {cos}^{2}x}$

we get

$\displaystyle\rm \: \: = \: \int\dfrac{\dfrac{1}{ {cos}^{2}x}}{ \dfrac{1}{ {cos}^{2}x } \times {cos}^{ \frac{3}{7}}x \: {sin}^{ \frac{11}{7}}x} \: dx$

$\displaystyle\rm \: \: = \: \int\dfrac{ {sec}^{2} x}{ {cos}^{ \frac{3}{7} - 2}x \: {sin}^{ \frac{11}{7}}x} \: dx$

$\displaystyle\rm \: \: = \: \int\dfrac{ {sec}^{2} x}{ {cos}^{ \frac{3 - 14}{7}}x \: {sin}^{ \frac{11}{7}}x} \: dx$

$\displaystyle\rm \: \: = \: \int\dfrac{ {sec}^{2} x}{ {cos}^{ \frac{- 11}{7}}x \: {sin}^{ \frac{11}{7}}x} \: dx$

$\displaystyle\rm \: \: = \: \int\dfrac{ {sec}^{2} x}{ \: {tan}^{ \frac{11}{7}}x} \: dx$

Now, we use method of Substitution,

$\red{\rm :\longmapsto\:Put \: tanx = y} \\ \red{\rm :\longmapsto\: {sec}^{2}x \: dx = dy}$

$\displaystyle\rm \: \: = \: \int\dfrac{dy}{ \: {y}^{ \frac{11}{7}}} \:$

$\displaystyle\rm \: \: = \: \int {y}^{ - \frac{11}{7} }dy$

$\rm \: \: = \: \dfrac{ {y}^{ - \frac{11}{7} + 1} }{ - \frac{11}{7} + 1} + c$

$\: \: \: \: \: \: \: \red{\bigg \{ \because \: \displaystyle\rm \: \: \: \int \: {x}^{n}dx = \dfrac{ {x}^{n + 1} }{n + 1} + c \bigg \}}$

$\rm \: \: = \: \dfrac{ {y}^{\frac{ - 11 + 7}{7}} }{\frac{ - 11 + 7}{7}} + c$

$\rm \: \: = \: \dfrac{ {y}^{\frac{ -4}{7}} }{\frac{ - 4}{7}} + c$

$\rm \: \: = \: - \: \dfrac{7}{4} \times \dfrac{1}{ {y}^{ \frac{4}{7} } } + c$

$\rm \: \: = \: - \: \dfrac{7}{4} \times \dfrac{1}{ {tan}^{ \frac{4}{7} }x } + c$

$\rm \: \: = \: - \: \dfrac{7}{4} {cot}^{ \frac{4}{7} }x + c$

Additional Information :-

$\green{\boxed{ \bf{ \: \int \: sinx \: dx = - cosx + c}}}$

$\green{\boxed{ \bf{ \: \int \: cosx \: dx = sinx + c}}}$

$\green{\boxed{ \bf{ \: \int \: cotx \: dx= log |sinx| + c}}}$

$\green{\boxed{ \bf{ \: \int \: tanx \: dx = log |secx| + c}}}$

$\green{\boxed{ \bf{ \: \int \: secx \: dx = log |secx + tanx| + c}}}$

$\green{\boxed{ \bf{ \: \int \: cosecx dx= log |cosecx - cotx| + c}}}$

$\green{\boxed{ \bf{ \: \int \: {sec}^{2}x \: dx = tanx + c }}}$

$\green{\boxed{ \bf{ \: \int \: {cosec}^{2}x \: dx = - cotx + c }}}$