MHT CET · Maths · Indefinite Integration
\(\int\left(1+x-\frac{1}{x}\right) \mathrm{e}^{x+\frac{1}{x}} \mathrm{~d} x\) equal to
- A \((x+1) \mathrm{e}^{x+\frac{1}{x}}+\mathrm{c}\), (where c is a constant of integration)
- B \(-x \mathrm{e}^{x+\frac{1}{x}}+\mathrm{c}\), (where c is a constant of integration)
- C \(\quad(x-1) \mathrm{e}^{x+\frac{1}{x}}+\mathrm{c}\), (where c is a constant of integration)
- D \(x \mathrm{e}^{x+\frac{1}{x}}+\mathrm{c}\), (where c is a constant of integration)
Answer & Solution
Correct Answer
(D) \(x \mathrm{e}^{x+\frac{1}{x}}+\mathrm{c}\), (where c is a constant of integration)
Step-by-step Solution
Detailed explanation
Note that \(\int\left[x \mathrm{f}^{\prime}(x)+\mathrm{f}(x)\right] \mathrm{d} x=x \mathrm{f}(x)+\mathrm{c}\)
\(\begin{aligned}
\therefore \quad & \int\left(1+x-\frac{1}{x}\right) \mathrm{e}^{x+\frac{1}{x}} \mathrm{~d} x \\
& =\int\left[x \mathrm{e}^{x+\frac{1}{x}}\left(1-\frac{1}{x^2}\right)+\mathrm{e}^{x+\frac{1}{x}}\right] \mathrm{d} x \\
& =x \mathrm{e}^{x+\frac{1}{x}}+\mathrm{c}
\end{aligned}\)
\(\begin{aligned}
\therefore \quad & \int\left(1+x-\frac{1}{x}\right) \mathrm{e}^{x+\frac{1}{x}} \mathrm{~d} x \\
& =\int\left[x \mathrm{e}^{x+\frac{1}{x}}\left(1-\frac{1}{x^2}\right)+\mathrm{e}^{x+\frac{1}{x}}\right] \mathrm{d} x \\
& =x \mathrm{e}^{x+\frac{1}{x}}+\mathrm{c}
\end{aligned}\)
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
- The function \(f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}\) is increasing, ifMHT CET 2021 Medium
- If the angle between the vectors \(\bar{a}=2 \lambda^2 \hat{i}+4 \lambda \hat{j}+\hat{k}\) and \(\overline{\mathrm{b}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}\) is obtuse, then \(\lambda \in\)MHT CET 2021 Easy
- The growth of population is proportional to the number present. If the population of a colony doubles is 50 years, then the population will become triple in _____ yearsMHT CET 2020 Medium
- The particular solution of the differential equation \(\frac{d y}{d x}=e^{2 y} \cos x\), when \(y\left(\frac{\pi}{6}\right)=0\) isMHT CET 2022 Easy
- The parametric equations of the curve \(x^2+y^2+a x+b y=0\) areMHT CET 2023 Hard
- If \(\bar{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=-\hat{i}+2 \hat{j}-4 \hat{k}\) and \(\bar{c}=\hat{i}+\hat{j}-2 \hat{k}\), then \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=\)MHT CET 2021 Easy
More PYQs from MHT CET
- Two cards are drawn simultaneously from a well shuffled pack of 52 cards. If \(X\) is the random variable of getting queens, then the value of \(2 E(X)+3 E\left(X^2\right)\) for the number of queens isMHT CET 2025 Medium
- Which one of the following is NOT a bioherbicide of bacterial origin?MHT CET 2025 Easy
- What is the IUPAC name of \(\left(\mathrm{CH}_3\right)_2-\mathrm{N}-\mathrm{CH}_3\) ?MHT CET 2022 Easy
- A glass cube of length \(24 \mathrm{~cm}\) has a small air bubble trapped inside. When viewed normally from one face it is \(10 \mathrm{~cm}\) below the surface. When viewed normally from the opposite face, its apparent distance is \(6 \mathrm{~cm}\). The refractive index of glass isMHT CET 2021 Easy
- Which from following pairs is an example of isotones?MHT CET 2024 Easy
- The negation of the statement pattern \(\sim \mathrm{S} \vee(\sim \mathrm{r} \wedge \mathrm{s})\) is equivalent toMHT CET 2023 Easy