MHT CET · Maths · Indefinite Integration
If \(\int x^{x}(1+\log x) d x=k x^{x}+c\), then \(k=\)
- A \(\log _{e} e\)
- B \(\log _{e}\left(\frac{1}{e^{2}}\right)\)
- C \(\log _{e}\left(e^{2}\right)\)
- D \(\log _{e}\left(\frac{1}{e}\right)\)
Answer & Solution
Correct Answer
(A) \(\log _{e} e\)
Step-by-step Solution
Detailed explanation
Let \(I=\int x^{x}(1+\log x) d x\)
Put \(x^{x}=t \Rightarrow e^{x \log x}=t \Rightarrow e^{x \log x}(1+\log x)\) \(d x=d t\) \(\therefore x^{x}(1+\log x) d x=d t\)
Thus, \(I=\int d t=t+c\)
\(
I=x^{x}+c
\)
Comparing with given data, we write \(k=1=\log _{e} e\)
Put \(x^{x}=t \Rightarrow e^{x \log x}=t \Rightarrow e^{x \log x}(1+\log x)\) \(d x=d t\) \(\therefore x^{x}(1+\log x) d x=d t\)
Thus, \(I=\int d t=t+c\)
\(
I=x^{x}+c
\)
Comparing with given data, we write \(k=1=\log _{e} e\)
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
- Let \(\vec{v}=2 \hat{i}+2 \hat{j}-\hat{k}\) and \(\bar{w}=\hat{i}+3 \hat{k}\). If \(\bar{u}\) is a unit vector, then the maximum value of the scalar triple product \([\overline{\mathrm{u}} \overline{\mathrm{v}} \overline{\mathrm{W}}]\) isMHT CET 2021 Medium
- A spherical rain drop evaporates at a rate proportional to its surface area. If initially its radius is 3 mm and after 1 second it is reduced to 2 mm , then at any time t its radius is (where \(0 \leq \mathrm{t} \lt 3\) )MHT CET 2024 Hard
- \(\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1+\cos x}\) is equal toMHT CET 2022 Easy
- If \(x=\sqrt{\mathrm{e}^{\sin ^{-1} \mathrm{t}}}\) and \(y=\sqrt{\mathrm{e}^{\cos ^{-1} \mathrm{t}}}\), then \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\) isMHT CET 2023 Medium
- The equation of the plane passing through the line of intersection of the planes \(x+y+z=1\) and \(3 x+4 y+5 z=2\) and perpendicular to the XY- plane isMHT CET 2025 Medium
- The angle between the lines \(x=\mathrm{y}, \mathrm{z}=0\) and \(\mathrm{y}=0, \mathrm{z}=0\) isMHT CET 2025 Easy
More PYQs from MHT CET
- \(\mathrm{f}(x)=(\cos x+\mathrm{i} \sin x) \cdot(\cos 3 x+\mathrm{i} \sin 3 x) \cdots \cdots\)\([\cos (2 \mathrm{n}-1) x+\mathrm{i} \sin (2 \mathrm{n}-1) x], \mathrm{n} \in \mathbb{N} \text { Then }\) \(\mathrm{f}^{\prime \prime}(x)\) \(=,(\text {Where } i=\sqrt{-1})\)MHT CET 2025 Medium
- Calculate amount of methane formed by liberation of \(149.6 \mathrm{~kJ}\) of heat using following equation.
\( \mathrm{C}_{(\mathrm{s})}+2 \mathrm{H}_{2(\mathrm{~g})} \longrightarrow \mathrm{CH}_{4(\mathrm{~g})} \quad \Delta \mathrm{H}\) \(=-74.8 \mathrm{~kJ} / \mathrm{mol} \)MHT CET 2023 Medium - What is the moment of inertia of the electron moving in second Bohr orbit of hydrogen atom?
[ \(\mathrm{h}=\) Planck's constant, \(\mathrm{m}=\) mass of electron, \(\varepsilon_0=\) permittivity of free space, \(\mathrm{e}=\) charge on electron]MHT CET 2023 Medium - If one of the lines given by \(k x^2+x y-y^2=0\) bisect the angle between the co-ordinate axes, then the values of \(\mathrm{k}\) areMHT CET 2021 Medium
- What is the formal charge on carbon atom in \(\mathrm{CO}_3^{2-}\) ion?MHT CET 2021 Medium
- When Zn is treated with excess of \(\mathrm{NaOH}\), the product obtained isMHT CET 2011 Medium