MHT CET · Maths · Definite Integration
The value of \(\int \cos \left(\log _e(x)\right) \mathrm{d} x\) is equal to (where \(C\) is a constant of integration.)
- A \(x[\cos (\log x)-\sin (\log x)]+C\)
- B \(\frac{x}{2}[\sin (\log x)-\cos (\log x)]+C\)
- C \(\frac{x}{2}[\sin (\log x)+\cos (\log x)]+C\)
- D \(x[\cos (\log x)+\sin (\log x)]+C\)
Answer & Solution
Correct Answer
(C) \(\frac{x}{2}[\sin (\log x)+\cos (\log x)]+C\)
Step-by-step Solution
Detailed explanation
\( \int \cos \left(\log _{\mathrm{e}} x\right) \mathrm{d} x \operatorname{let}_{\log _{\mathrm{e}} x=t} \)
\( \Rightarrow \mathrm{d} x=e^t \)
\( \Rightarrow I=\int \cos t \cdot e^t \mathrm{~d} t=\cos t \cdot e^t+\sin t \cdot e^t\) \(-I [\text { Integrating by parts] } \)
\( \Rightarrow 2 I=e^t(\cos t+\sin t) \)
\( \Rightarrow I=\frac{x}{2}\left\{\cos \left(\log _{\mathrm{e}} x\right)+\sin \left(\log _{\mathrm{e}} x\right)\right\}\)
\( \Rightarrow \mathrm{d} x=e^t \)
\( \Rightarrow I=\int \cos t \cdot e^t \mathrm{~d} t=\cos t \cdot e^t+\sin t \cdot e^t\) \(-I [\text { Integrating by parts] } \)
\( \Rightarrow 2 I=e^t(\cos t+\sin t) \)
\( \Rightarrow I=\frac{x}{2}\left\{\cos \left(\log _{\mathrm{e}} x\right)+\sin \left(\log _{\mathrm{e}} x\right)\right\}\)
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
- The solution of the differential equation \(\log \left(\frac{\mathrm{dy}}{\mathrm{d} x}\right)=9 x-6 \mathrm{y}+6\) is
(given that \(\mathrm{y}=1\) when \(x=0\) )MHT CET 2020 Medium - If the surrounding air is kept at \(25^{\circ} \mathrm{C}\) and a body cools from \(80^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) in 30 minutes, then temperature of the body after one hour will beMHT CET 2021 Hard
- If \(\varphi^{\prime}\) is the angle between the lines \(a x^{2}+2 h x y+b y^{2}=0\), then angle between \(x^{2}+2 x y \sec \theta+y^{2}=0\) isMHT CET 2009 Medium
- The parametric representation of a point on the ellipse whose foci are \((-1,0)\) and \((7,0)\) and eccentricity \(1 / 2\), isMHT CET 2007 Easy
- Eight chairs are numbered 1 to 8 . Two women and three men wish to occupy one chair each. First the women choose chairs from amongst the chairs marked 1 to 4 , and then the men select the chairs from amongst the remaining. The number of possible arrangements isMHT CET 2024 Medium
- If \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors and \(\bar{p}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{q}=\frac{\bar{c} \times \bar{a}}{[a} \bar{b} \overline{c]}, \bar{r}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}\),
then \(\bar{a} \cdot \bar{p}+\bar{b} \cdot \bar{q}+\bar{c} \cdot \bar{r}=\)MHT CET 2020 Easy
More PYQs from MHT CET
- A particle executes the simple hormonic motion with an amplitude ‘ ’. The distance travelled by it in one periodic time isMHT CET 2019 Medium
- 1-chlorobutane on reaction with alcoholic potash givesMHT CET 2007 Easy
- Two solid spheres of radii \(R_1\) and \(R_2\) made of same material and have similar surfaces. The spheres are raised to the same temperature and then allowed to cool under identical conditions. Assuming spheres to be perfect conductors of heat, then the ratio of initial rates of cooling areMHT CET 2022 Easy
- According to Raoult's law mole fraction of solute in solution is given by formulaMHT CET 2021 Easy
- Which of the following organic compounds could not be dried by anhydrous \(\mathrm{CaCl}_{2}\) ?MHT CET 2012 Hard
- In a triangle \(\mathrm{ABC}\) with usual notations, if \(\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\), then area ofMHT CET 2020 Medium