If I(x) = e^ ^2x x( 2x - x) dx and I(0) = 1, then I ( 3) is equal to:

Given: I(x) = e^ ^2x x( 2x - x) dx and I(0) = 1. Find: I ( 3). Use 2x = 2 x x to rewrite the integrand: I(x)= e^ ^2x x (2 x x- x) dx = e^ ^2x x x (2 x-1) dx Observe that ddx (e^ ^2x x)=e^ ^2x(2 x x)( x)-e^ ^2x x =e^ ^2x x(2 x x- x) which is exactly the given integrand. Therefore, I(x)=e^ ^2x x+C Now apply the initial condition I(0)=1: 1=e^ ^2 0 0+C=1 1+C So, C=0 Hence, I(x)=e^ ^2x x Evaluate at x= 3: I ( 3)=e^ ^2 ( 3) ( 3) Using 3= 32 and 3= 12, I ( 3)=e^ ( 32)^2 12=e^ 34 12= e^ 342 Therefore, the correct option is C. The solution heading states The Correct Option is D, but the worked steps conclude I ( 3)= e^ 342 which matches option C exactly. The final computed value is therefore taken as authoritative, so the answer is C.

If I(x) = e^ ^2x x( 2x - x) dx and I(0) = 1, then I ( 3) is equal… — Solution | JEE Mathematics

MCQMediumJEE 2023Indefinite Integrals

If $I(x) = \int e^{\sin^2x}\cos x(\sin 2x - \sin x) \, dx$ and $I(0) = 1$ , then $I\left(\frac{\pi}{3}\right)$ is equal to:

A
$e^{3/4}$
B
$-e^{3/4}$
C
$\frac{1}{2}e^{3/4}$
D
$-\frac{1}{2}e^{3/4}$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: $I(x) = \int e^{\sin^2x}\cos x(\sin 2x - \sin x) \, dx$ and $I(0) = 1$ .

Find: $I\left(\frac{\pi}{3}\right)$ .

Use $\sin 2x = 2\sin x \cos x$ to rewrite the integrand:

I(x)=\int e^{\sin^2x}\cos x\,(2\sin x\cos x-\sin x)\,dx

=\int e^{\sin^2x}\sin x\cos x\,(2\cos x-1)\,dx

Observe that

\frac{d}{dx}\left(e^{\sin^2x}\cos x\right)=e^{\sin^2x}(2\sin x\cos x)(\cos x)-e^{\sin^2x}\sin x

=e^{\sin^2x}\cos x(2\sin x\cos x-\sin x)

which is exactly the given integrand.

Therefore,

I(x)=e^{\sin^2x}\cos x+C

Now apply the initial condition $I(0)=1$ :

1=e^{\sin^2 0}\cos 0+C=1\cdot 1+C

So,

C=0

Hence,

I(x)=e^{\sin^2x}\cos x

Evaluate at $x=\frac{\pi}{3}$ :

I\left(\frac{\pi}{3}\right)=e^{\sin^2\left(\frac{\pi}{3}\right)}\cos\left(\frac{\pi}{3}\right)

Using $\sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}$ and $\cos \frac{\pi}{3}=\frac{1}{2}$ ,

I\left(\frac{\pi}{3}\right)=e^{\left(\frac{\sqrt{3}}{2}\right)^2}\cdot \frac{1}{2}=e^{\frac{3}{4}}\cdot \frac{1}{2}= \frac{e^{\frac{3}{4}}}{2}

Therefore, the correct option is C.

Answer Discrepancy Note

The solution heading states The Correct Option is D, but the worked steps conclude

I\left(\frac{\pi}{3}\right)=\frac{e^{\frac{3}{4}}}{2}

which matches option C exactly. The final computed value is therefore taken as authoritative, so the answer is C.

Common mistakes

Using the heading Option D without checking the worked solution is incorrect because the actual computation gives $\frac{e^{3/4}}{2}$ . Always trust the derived result over a mismatched label.
Expanding $\sin 2x$ incorrectly is a common error. The correct identity is $\sin 2x = 2\sin x \cos x$ , and using a wrong identity changes the integrand completely.
Forgetting to apply the initial condition $I(0)=1$ leaves an arbitrary constant $C$ . After finding the antiderivative, always substitute the given value to determine $C$ .

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