KCET · Maths · Indefinite Integration
The value of \(\int e^{\sin x} \sin 2 x d x\) is
- A \(2 e^{\sin x}(\sin x-1)+C\)
- B \(2 e^{\sin x}(\sin x+1)+C\)
- C \(2 e^{\sin x}(\cos x+1)+C\)
- D \(2 e^{\sin x}(\cos x-1)+C\)
Answer & Solution
Correct Answer
(A) \(2 e^{\sin x}(\sin x-1)+C\)
Step-by-step Solution
Detailed explanation
We have, \(\int e^{\sin x} \sin 2 x d x\)
\(\begin{aligned}
&=\int e^{\sin x} \cdot(2 \sin x \cos x) d x \\
&=2 \int e^{\sin x} \sin x \cos x d x
\end{aligned}\)
let \(\sin x=t\)
\(\begin{aligned}
\Rightarrow \cos x d x &=d t=2 \int e^{t} \cdot t d t \\
&=2\left[t \int e^{t} d t-\int\left(\frac{d}{d t}(t) \int e^{t} d t\right) d t\right] \\
&=2\left[t \cdot e^{t}-\int e^{t} d t\right] \\
&=2\left[t \cdot e^{t}-e^{t}\right]+C \\
&=2\left[e^{t}(t-1)\right]+C \\
&=2\left[e^{\sin x}(\sin x-1)\right]+C \\
&=2 e^{\sin x}(\sin x-1)+C
\end{aligned}\)
\(\begin{aligned}
&=\int e^{\sin x} \cdot(2 \sin x \cos x) d x \\
&=2 \int e^{\sin x} \sin x \cos x d x
\end{aligned}\)
let \(\sin x=t\)
\(\begin{aligned}
\Rightarrow \cos x d x &=d t=2 \int e^{t} \cdot t d t \\
&=2\left[t \int e^{t} d t-\int\left(\frac{d}{d t}(t) \int e^{t} d t\right) d t\right] \\
&=2\left[t \cdot e^{t}-\int e^{t} d t\right] \\
&=2\left[t \cdot e^{t}-e^{t}\right]+C \\
&=2\left[e^{t}(t-1)\right]+C \\
&=2\left[e^{\sin x}(\sin x-1)\right]+C \\
&=2 e^{\sin x}(\sin x-1)+C
\end{aligned}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \( 3 \tan ^{-1} x+\cot ^{-1}=\pi \) then \( x \) equal toKCET 2016 Easy
- If ' \(n\) ' is a positive integer, then \(\mathrm{n}^{3}+2 \mathrm{n}\) is divisible byKCET 2009 Easy
- \(2 \cos ^{-1} x=\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)\) is valid for all values of \(x\) satisfyingKCET 2012 Medium
- If \(P\) is a point on \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with focii \(S\) and \(S^{\prime}\), then the maximum value of \(\triangle S P S^{\prime}\) isKCET 2011 Medium
- A gardener is digging a plot of land. As he gets tired, he works more slowly. After 't' minutes he
is digging at a rate of \( \frac{2}{\sqrt{t}} \) square metres per minute. How long will it take him to dig an area of
\( 40 \) square metres?KCET 2014 Easy - If the set \(x\) contains 7 elements and set \(y\) contains 8 elements, then the number of bijections from \(x\) to \(y\) isKCET 2022 Easy
More PYQs from KCET
- The type of antibodies produced during the allergic reactionKCET 2022 Easy
- The value of the \( \sin 1^{\circ}+\sin 2^{\circ}+\ldots \ldots+\sin 359^{\circ} \) is equal toKCET 2016 Medium
- The total number of terms in the expansion of \( (x+a)^{47}-(x-a)^{47} \) after simplification isKCET 2017 Easy
- If \([x]^2-5[x]+6=0\), where \([x]\) denotes the greatest integer function, thenKCET 2024 Easy
- If \( f(x)=\left\{\begin{aligned} K x^{3} ; & \text { if } x \leq 2 \\ 3 ; & \text { if } x>2 \end{aligned}\right. \) is continuous at \( x=2 \), then the value of \( K \) isKCET 2017 Easy
- The following factors indicate improved reproductive health of the society. Choose the correct option.
A. Better detection and cure of disease
B. Better post-natal care
C. Medically assisted deliveries
D. Increased MMR
Select the code for the correct answer from the options given below.KCET 2018 Medium