JEE Mains · Maths · STD 12 - 7.1 indefinite integral
Let \(\int x^3 \sin x \mathrm{~d} x=g(x)+C\), where \(C\) is the constant of integration. If \(8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^3+\beta \pi^2+\gamma, \alpha, \beta, \gamma \in Z\), then \(\alpha+\beta-\gamma\) equals :
- A 48
- B 55
- C 62
- D 47
Answer & Solution
Correct Answer
(B) 55
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int x^3 \sin x d x=-x^3 \cos x+\int 3 x^2 \cos x d x \\ & =-x^3 \cos x+3 x^2 \sin x-\int 6 x \sin x d x \\ & =-x^3 \cos x+3 x^2 \sin x+6 x \cos x-6 \sin x+c \end{aligned}\) So \(g(x)=-x^3 \cos x+3 x^2 \sin x+6 x \cos x-6 \sin x\)…
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 \((\sqrt{3}+\mathrm{i})^{100}=2^{99}(\mathrm{p}+\mathrm{i} \mathrm{q})\), then \(\mathrm{p}\) and \(\mathrm{q}\) are roots of the equation :JEE Mains 2021 Hard
- Let \(y = y ( x )\) be the solution of the differential equation \(x d y-y d x=\sqrt{\left(x^{2}-y^{2}\right)} d x, x \geq 1\), with \(y (1)=0 .\) If the area bounded by the line \(x =1, x = e ^{\pi}, y =0\) and \(y = y ( x )\) is \(\alpha e ^{2 \pi}+\beta\) then the value of \(10(\alpha+\beta)\) is equal to ....... .JEE Mains 2021 Medium
- The plane which bisects the line segment joining the points \((-3, -3, 4)\) and \((3, 7, 6)\) at right angles, passes through which one of the following points?JEE Mains 2019 Hard
- Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines \(x + (k-1)y + 3 = 0\) and \(2x + k^2 y - 4 = 0\). If the line \(x - y + 2 = 0\) intersects the circle at the points A and B, then \((AB)^2\) is equal to:JEE Mains 2026 Hard
- If the term independent of \(x\) in the expansion of \(\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}\) is 105 , then \(\mathrm{a}^2\) is equal to :JEE Mains 2024 Medium
- The set of all real values of \(\lambda \) for which exactly two common tangents can be drawn to the circles \(x^2 + y^2 - 4x - 4y+ 6\, = 0\) and \(x^2 + y^2 - 10x - 10y + \lambda \, = 0\) is the interval:JEE Mains 2014 Hard
More PYQs from JEE Mains
- Let \(A = \begin{bmatrix} 1 & 2 \\ 1 & \alpha \end{bmatrix}\) and \(B = \begin{bmatrix} 3 & 3 \\ \beta & 2 \end{bmatrix}\). If \(A^2 - 4A + I = O\) and \(B^2 - 5B - 6I = O\), then among the two statements :
(S1): \([(B-A)(B+A)]^T = \begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix}\)
and
(S2): \(\det(\text{adj}(A+B)) = -5\),JEE Mains 2026 Hard - Let the set of all values of r, for which the circles \((x+1)^{2}+(y+4)^{2}=r^{2}\) and \(x^{2}+y^{2}-4x-2y-4=0\) intersect at two distinct points be the interval \((\alpha, \beta)\). Then \(\alpha\beta\) is equal toJEE Mains 2026 Easy
- The sum of the first \(20\) terms common between the series \(3 +7 + 1 1 + 15+ ... ......\) and \(1 +6+ 11 + 16+ ......\), isJEE Mains 2014 Hard
- If \(\mathrm{U}_{\mathrm{n}}=\left(1+\frac{1}{\mathrm{n}^{2}}\right)\left(1+\frac{2^{2}}{\mathrm{n}^{2}}\right)^{2} \ldots\left(1+\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}}\right)^{\mathrm{n}}\), then \(\lim _{n \rightarrow \infty}\left(U_{n}\right)^{\frac{-4}{n^{2}}}\) is equal to :JEE Mains 2021 Hard
- For \(x \in R\), two real valued functions \(f(x)\) and \(g(x)\) are such that, \(g(x)=\sqrt{x}+1\) and \(f o g(x)=x+3-\sqrt{x}\). Then \(f(0)\) is equal toJEE Mains 2023 Hard
- Let \(f\) and \(g\) be twice differentiable functions on \(R\) such that \(f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x\) \(f^{\prime}(1)=4 g^{\prime}(1)-3=9\) \(f(2)=3 g(2)=12\) Then which of the following is NOT true ?JEE Mains 2023 Hard