MHT CET · Maths · Definite Integration
\(
\int_{-a}^{a} x^{2}\left(\frac{e^{x^{3}}-e^{-x^{3}}}{e^{x^{3}}+e^{-x^{3}}}\right) d x=
\)
- A \(a^{2}\)
- B 0
- C \(a\)
- D \(2 \int_{0}^{a} x^{2}\left(\frac{e^{x^{3}}-e^{-x^{3}}}{e^{x^{3}}+e^{-x^{3}}}\right) d x\)
Answer & Solution
Correct Answer
(B) 0
Step-by-step Solution
Detailed explanation
Let
\(f(x) =x^{2}\left[\frac{e^{x^{3}}-e^{-x^{3}}}{e^{x^{3}}+e^{-x^{3}}}\right]=x^{2}\left[\frac{e^{x^{3}}-\frac{1}{e^{x^{3}}}}{e^{x^{3}}+\frac{1}{e^{x^{3}}}}\right]=\) \(x^{2}\left[\frac{\left(e^{x^{3}}\right)^{2}-1}{\left(e^{x^{3}}\right)^{2}+1}\right] \)
\( f(-x) =(-x)^{2}\left[\frac{e^{-x^{3}}-e^{x^{3}}}{e^{-x^{3}}+e^{x^{3}}}\right]\)
\(=x^{2}\left[\frac{\frac{1}{e^{x^{3}}}-e^{x^{3}}}{\frac{1}{e^{x^{3}}}+e^{x^{3}}}\right]=x^{2}\left[\frac{1-\left(e^{x^{3}}\right)^{2}}{1+\left(e^{x^{3}}\right)^{2}}\right]=\) \(-x^{2}\left[\frac{\left(e^{x^{3}}\right)^{2}-1}{1+\left(e^{x^{3}}\right) 2}\right]=-f(x)\)
Thus \(f(-x)=-f(x) \Rightarrow\) Given function is an odd function.
\(\therefore \quad I=0\)
\(f(x) =x^{2}\left[\frac{e^{x^{3}}-e^{-x^{3}}}{e^{x^{3}}+e^{-x^{3}}}\right]=x^{2}\left[\frac{e^{x^{3}}-\frac{1}{e^{x^{3}}}}{e^{x^{3}}+\frac{1}{e^{x^{3}}}}\right]=\) \(x^{2}\left[\frac{\left(e^{x^{3}}\right)^{2}-1}{\left(e^{x^{3}}\right)^{2}+1}\right] \)
\( f(-x) =(-x)^{2}\left[\frac{e^{-x^{3}}-e^{x^{3}}}{e^{-x^{3}}+e^{x^{3}}}\right]\)
\(=x^{2}\left[\frac{\frac{1}{e^{x^{3}}}-e^{x^{3}}}{\frac{1}{e^{x^{3}}}+e^{x^{3}}}\right]=x^{2}\left[\frac{1-\left(e^{x^{3}}\right)^{2}}{1+\left(e^{x^{3}}\right)^{2}}\right]=\) \(-x^{2}\left[\frac{\left(e^{x^{3}}\right)^{2}-1}{1+\left(e^{x^{3}}\right) 2}\right]=-f(x)\)
Thus \(f(-x)=-f(x) \Rightarrow\) Given function is an odd function.
\(\therefore \quad I=0\)
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 \(\mathrm{y}+\frac{\mathrm{d}}{\mathrm{d} x}(x \mathrm{y})=x(\sin x+\log x)\) thenMHT CET 2025 Medium
- A vector \(\overline{\mathrm{a}}\) has components 1 and \(2 \mathrm{p}\) with respect to a rectangular Cartesian system. This system is rotated through a certain angle about origin in the counter clock wise sense. If, with respect to the new system, \(\bar{a}\) has components 1 and \((p+1)\), thenMHT CET 2023 Hard
- The raw data \(x_1, x_2, \ldots \ldots, x_{\mathrm{n}}\) is an A.P. with common difference \(\mathrm{d}\) and first term \(0 . \bar{x}\) and \(\sigma^2\) are mean and variance of \(x_{\mathrm{i}}, \mathrm{i}=1,2, \ldots \ldots \mathrm{n}\), then \(\sigma^2\) isMHT CET 2023 Medium
- If \((p \wedge \sim r) \rightarrow(\sim p \vee q)\) has truth value False, then truth values of \(p, q, r\) are respectively.MHT CET 2024 Easy
- The area of the triangle formed by the lines joining the vertex of the parabola \(x^2=20 \mathrm{y}\) to the end of its Latus rectum isMHT CET 2025 Medium
- A die is thrown 100 times, then the standard deviation of getting an even number isMHT CET 2020 Medium
More PYQs from MHT CET
- The equivalent form of the statement is ________MHT CET 2019 Easy
- Two satellites A and B rotate around a planet's orbit having radii \(4 \mathrm{R}\) and \(\mathrm{R}\) respectively. If the speed of satellite \(\mathrm{A}\) is \(3 \mathrm{~V}\) then the speed of satellite B isMHT CET 2022 Easy
- If an electron in a hydrogen atom jumps from an orbit of level \(\mathrm{n}=3\) to orbit of level \(\mathrm{n}=2\), then the emitted radiation frequency is (where \(\mathrm{R}=\) Rydberg's constant, \(\mathrm{C}=\) Velocity of light)MHT CET 2023 Medium
- A capillary tube stands with its lower end dipped into liquid for which the angle of contact is \(90^{\circ}\). The liquid willMHT CET 2024 Easy
- What is the radius of fourth orbit of \(\mathrm{Be}^{+++}\)?MHT CET 2023 Easy
- \(\int \frac{x^{e-1}+e^{x-1}}{x^{e}+e^{x}} d x\) is equal toMHT CET 2008 Medium