\int e^\sec x(\sec x\tan xf(x) + \sec x\tan x + sex^2x)dx = esecxf(x) + C then a possible choice of f(x) is :-

Indefinite Integrals PYQ — JEE Main 2019 · 9 Apr · Shift 2 · Q35 with Solution

View written solutionFree

Correct answer: D

We need to interpret the given integral carefully.

The expression appears to mean

\int e^{\sec x}\Big(\sec x\tan x\,f(x)+\sec x\tan x+\sec^2 x\Big)\,dx = e^{\sec x}f(x)+C.

So the integrand must be the derivative of $e^{\sec x}f(x)$ .

1. Differentiate the RHS

Using the product rule,

\frac{d}{dx}\big(e^{\sec x}f(x)\big) = e^{\sec x}\cdot \frac{d}{dx}(\sec x)\,f(x)+e^{\sec x}f'(x).

Since

\frac{d}{dx}(\sec x)=\sec x\tan x,

we get

\frac{d}{dx}\big(e^{\sec x}f(x)\big) = e^{\sec x}\Big(\sec x\tan x\,f(x)+f'(x)\Big).

This must equal the given integrand:

e^{\sec x}\Big(\sec x\tan x\,f(x)+\sec x\tan x+\sec^2 x\Big).

Hence,

f'(x)=\sec x\tan x+\sec^2 x.

2. Integrate to find $f(x)$

Now,

f(x)=\int (\sec x\tan x+\sec^2 x)\,dx.

Using standard results,

\int \sec x\tan x\,dx=\sec x, \qquad \int \sec^2 x\,dx=\tan x.

Therefore,

f(x)=\sec x+\tan x+C.

So a possible choice is any function of the form

\sec x+\tan x+\text{constant}.

3. Match with the options

Check the options:

A: $x\sec x+\tan x+\frac12$
Derivative is not $\sec x\tan x+\sec^2 x$ .
B: $\sec x+x\tan x-\frac12$
Derivative is not $\sec x\tan x+\sec^2 x$ .
C: $\sec x-\tan x-\frac12$
Derivative is $\sec x\tan x-\sec^2 x$ , not correct.
D: $\sec x+\tan x+\frac12$
Derivative is
$\sec x\tan x+\sec^2 x,$
which matches.

Hence the possible choice is

\boxed{\sec x+\tan x+\frac12}.

So Option D is correct.

4. Comparison with stored answer

Stored correct answer: D
Derived answer: D

They agree.

More from Indefinite Integrals

View written solutionFree

Correct answer: D

We need to interpret the given integral carefully.

The expression appears to mean

\int e^{\sec x}\Big(\sec x\tan x\,f(x)+\sec x\tan x+\sec^2 x\Big)\,dx = e^{\sec x}f(x)+C.

So the integrand must be the derivative of $e^{\sec x}f(x)$ .

1. Differentiate the RHS

Using the product rule,

\frac{d}{dx}\big(e^{\sec x}f(x)\big) = e^{\sec x}\cdot \frac{d}{dx}(\sec x)\,f(x)+e^{\sec x}f'(x).

Since

\frac{d}{dx}(\sec x)=\sec x\tan x,

we get

\frac{d}{dx}\big(e^{\sec x}f(x)\big) = e^{\sec x}\Big(\sec x\tan x\,f(x)+f'(x)\Big).

This must equal the given integrand:

e^{\sec x}\Big(\sec x\tan x\,f(x)+\sec x\tan x+\sec^2 x\Big).

Hence,

f'(x)=\sec x\tan x+\sec^2 x.

2. Integrate to find $f(x)$

Now,

f(x)=\int (\sec x\tan x+\sec^2 x)\,dx.

Using standard results,

\int \sec x\tan x\,dx=\sec x, \qquad \int \sec^2 x\,dx=\tan x.

Therefore,

f(x)=\sec x+\tan x+C.

So a possible choice is any function of the form

\sec x+\tan x+\text{constant}.

3. Match with the options

Check the options:

A: $x\sec x+\tan x+\frac12$
Derivative is not $\sec x\tan x+\sec^2 x$ .
B: $\sec x+x\tan x-\frac12$
Derivative is not $\sec x\tan x+\sec^2 x$ .
C: $\sec x-\tan x-\frac12$
Derivative is $\sec x\tan x-\sec^2 x$ , not correct.
D: $\sec x+\tan x+\frac12$
Derivative is
$\sec x\tan x+\sec^2 x,$
which matches.

Hence the possible choice is

\boxed{\sec x+\tan x+\frac12}.

So Option D is correct.

4. Comparison with stored answer

Stored correct answer: D
Derived answer: D

They agree.

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2. Integrate to find $f(x)$

3. Match with the options

4. Comparison with stored answer

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