Q. The integral value of the function $x sec^{2} x$ is

144 177 Integrals Report Error

A

$x tan - lo g ∣ cos x ∣ + C$

13%

B

$x tan x + lo g ∣ cos x ∣ + C$

35%

C

$x tan + lo g ∣ sec x ∣ + C$

30%

D

None of these

21%

Solution:

Let $I = \int x sec^{2} x d x$ .
On taking $x$ as first function and $sec^{2} x$ as second function and integrating by parts, we get
$I = x \int sec^{2} x d x - \int [\frac{d}{d x} (x) \int sec^{2} x d x] d x$
$= x tan x - \int tan x d x = x tan x - lo g ∣ sec x ∣ + C$
$\Rightarrow I = x tan x + lo g ∣ cos x ∣ + C$
$[lo g ∣ sec x ∣ = lo g ∣ ∣ \frac{1}{c o s x} ∣ ∣ = lo g 1 - lo g ∣ cos x ∣$
$= - lo g ∣ cos x ∣ (∵ lo g = 0)]$