The integral {² x}{( x + x)^{9/2}} \, dx equals (for some arbitraryconstant K) [JEE Adv 2012 P1]

Correct Answer is : −1(sec⁡x+tan⁡x)11/2{111+17(sec⁡x+tan⁡x)2}+K-{1}{( x + x)¹1/2}} ≤ft\{ 1/11 + 1/7 ( x + x)² \} + K−(secx+tanx)11/21{111+71(secx+tanx)2}+K

-{1}{( x + x)¹1/2}} ≤ft\{ 1/11 - 1/7 ( x + x)² \} + K

-{1}{( x + x)¹1/2}} ≤ft\{ 1/11 + 1/7 ( x + x)² \} + K

{1}{( x + x)¹1/2}} ≤ft\{ 1/11 + 1/7 ( x + x)² \} + K

The integral {² x}{( x + x)^{9/2}} \, dx equals (for some arbitraryconstant K) [JEE Adv 2012 P1]

Correct Answer is : −1(sec⁡x+tan⁡x)11/2{111+17(sec⁡x+tan⁡x)2}+K-{1}{( x + x)¹1/2}} ≤ft\{ 1/11 + 1/7 ( x + x)² \} + K−(secx+tanx)11/21{111+71(secx+tanx)2}+K

Test Index

Indefinite Integrals

Section: Mathematics

Question : 1 of 2

Marks: +1, -0

The integral

\int \frac{\sec^2 x}{(\sec x + \tan x)^{9/2}} \, dx

equals (for some arbitraryconstant K)

[JEE Adv 2012 P1]

$-\frac{1}{(\sec x + \tan x)^{11/2}} \left\{ \frac{1}{11} - \frac{1}{7} (\sec x + \tan x)^2 \right\} + K$
$-\frac{1}{(\sec x + \tan x)^{11/2}} \left\{ \frac{1}{11} - \frac{1}{7} (\sec x + \tan x)^2 \right\} + K$
$-\frac{1}{(\sec x + \tan x)^{11/2}} \left\{ \frac{1}{11} + \frac{1}{7} (\sec x + \tan x)^2 \right\} + K$
$\frac{1}{(\sec x + \tan x)^{11/2}} \left\{ \frac{1}{11} + \frac{1}{7} (\sec x + \tan x)^2 \right\} + K$

Validate

Solution:

I =

\int \frac{\sec^2 x}{(\sec x + \tan x)^{9/2}} \, dx

Let sec x + tan x = t ⇒sec x - tan x = 1 / t.

Now, (sec x tan x +

\sec^2

x)dx = dt

sec x(sec x + tan x)dx = dt

secx dx =

\frac{dt}{t}, \frac{1}{2} \left( t + \frac{1}{t} \right)

= sec x

I =

\frac{1}{2} \int \frac{\left(t + \frac{1}{t}\right) dt}{t^{9/2} t}

\frac{1}{2} \int (t^{-9/2} + t^{-13/2}) dt

\frac{1}{2} \left[ \frac{t^{-9/2+1}}{-\frac{9}{2}+1} + \frac{t^{-\frac{13}{21}}}{-\frac{13}{2}+1} \right]

\frac{1}{2} \left[ \frac{t^{-7/2}}{-\frac{7}{2}} + \frac{t^{-11/2}}{-\frac{11}{2}} \right]

-\frac{1}{7} t^{-7/2} - \frac{1}{11} t^{-11/2}

-\frac{1}{7} \frac{1}{t^{7/2}} - \frac{1}{11} \frac{1}{t^{11/2}}

-\frac{1}{t^{11/2}} \left( \frac{1}{11} + \frac{t^2}{7} \right)

-\frac{1}{(\sec x + \tan x)^{11/2}} \left\{ \frac{1}{11} + \frac{1}{7} (\sec x + \tan x)^2 \right\} + K

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