JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(\int_{0}^{\pi}\left(\sin ^{3} x\right) e^{-\sin ^{2} x} d x=\alpha-\frac{\beta}{e} \int_{0}^{1} \sqrt{t} e^{t} d t\), then \(\alpha+\beta\) is equal to \(....\)
- A \(4\)
- B \(5\)
- C \(6\)
- D \(7\)
Answer & Solution
Correct Answer
(B) \(5\)
Step-by-step Solution
Detailed explanation
\(I=2 \int_{0}^{\pi / 2} \sin ^{3} x e^{-\sin ^{2} x} d x\) \(=2 \int_{0}^{\pi / 2} \sin x e^{-\sin ^{2} x} d x+\int_{0}^{\pi / 2} \cos _{I} x \underbrace{e^{-\sin ^{2} x}(-\sin 2 x)}_{\text {II }} d x\)…
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 the tangent to the parabola \(y^2 = x\) at a point \(\left( {\alpha ,\beta } \right)\,,\,\left( {\beta > 0} \right)\) is also a tangent to the ellipse, \(x^2 + 2y^2 = 1\), then \(a\) is equal toJEE Mains 2019 Hard
- Let \(X\) be a random variable having binomial distribution \(B (7, p )\). If \(P ( X =3)=5 P ( X =4)\), then the sum of the mean and the variance of \(X\) isJEE Mains 2022 Medium
- Let \(\theta\) be the acute angle between the tangents to the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{1}=1\) and the circle \(x^{2}+y^{2}=3\) at their point of intersection in the first quadrant. Then \(\tan \theta\) is equal to :JEE Mains 2021 Hard
- Let \(f(x)=\sqrt{\lim _{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}\) be differentiable in \((-\infty, 0) \cup(0, \infty)\) and \(f(1)=1\). Then the value of \(ea\), such that \(f(a)=0\), is equal to ...........JEE Mains 2024 Hard
- Two fair dice are thrown. The numbers on them are taken as \(\lambda\) and \(\mu\), and a system of linear equations \(x+y+z=5\) ; \(x+2 y+3 z=\mu\) ; \(x+3 y+\lambda z=1\) is constructed. If \(\mathrm{p}\) is the probability that the system has a unique solution and \(\mathrm{q}\) is the probability that the system has no solution, then :JEE Mains 2021 Hard
- If a curve the \(y = f(x)\) passes through point \((1, -1)\) and satisfies the differential equation \(y\left( {1 + xy} \right)dx = xdy\) then \(f\left( { - \frac{1}{2}} \right) = \) . . . . .JEE Mains 2016 Hard
More PYQs from JEE Mains
- Let \( y=y(x) \) be the solution of the differential equation \( secx\frac{dy}{dx}-2y=2+3~sin~x, x\in(-\frac{\pi}{2},\frac{\pi}{2}), \) \( y(0)=-\frac{7}{4}. \) Then \( y(\frac{\pi}{6}) \) is equal to :JEE Mains 2026 Hard
- Let \(e\) be the base of natural logarithm and let \(f: \{1, 2, 3, 4\} \rightarrow \{1, e, e^2, e^3\}\) and \(g: \{1, e, e^2, e^3\} \rightarrow \left\{1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}\right\}\) be two bijective functions such that \(f\) is strictly decreasing and \(g\) is strictly increasing. If \(\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\dfrac{1}{2}\right)\right\}\right]^x\), then the area of the region \(R = \{(x, y): x^2 \leq y \leq \phi(x), 0 \leq x \leq 1\}\) is:JEE Mains 2026 Hard
- The integral \(\int_{\pi /6}^{\pi /4} {\frac{{dx}}{{\sin \,2x\,\left( {{{\tan }^5}\,x + {{\cot }^5}\,x} \right)}}} \) equalsJEE Mains 2019 Hard
- The number of real values \(\lambda\), such that the system of linear equations \(2 x-3 y+5 z=9\) ; \(x+3 y-z=-18\) ; \(3 x-y+\left(\lambda^{2}-1 \lambda \mid\right) z=16\) has no solution, is :-JEE Mains 2022 Hard
- If the function \(f:(-\infty,-1] \rightarrow(a, b]\) defined by \(f(x)=e^{x^3-3 x+1}\) is one-one and onto, then the distance of the point \(\mathrm{P}(2 \mathrm{~b}+4, \mathrm{a}+2)\) from the line \(x+e^{-3} y=4\) is :JEE Mains 2024 Hard
- Let \(f(\mathrm{x})=\mathrm{x}^5+2 \mathrm{e}^{\mathrm{x} / 4}\) for all \(\mathrm{x} \in \mathrm{R}\). Consider a function \(g(x)\) such that \((gof) (x)=x\) for all \(x \in R\). Then the value of \(8 g^{\prime}(2)\) is :JEE Mains 2024 Hard