JEE Mains · Maths · STD 12 - 7.1 indefinite integral
If \(\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _e\left|\tan \frac{x}{2}\right|+C\) where \(\alpha, \beta \in \mathbb{R}\) and \(\mathrm{C}\) is constant of integration , then the value of \(8(\alpha+\beta)\) equals ...........
- A \(5\)
- B \(1\)
- C \(6\)
- D \(45\)
Answer & Solution
Correct Answer
(B) \(1\)
Step-by-step Solution
Detailed explanation
\(\int \operatorname{cosec}^3 x \cdot \operatorname{cosec}^2 x d x=I\) By applying integration by parts \( I=-\cot x \operatorname{cosec}^3 x+\int \cot x\left(-3 \operatorname{cosec}^2 x \cot x \operatorname{cosec} x\right) d x \)…
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
- \(\lim\limits_{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{\frac{1}{x^{2}}}\) is equal toJEE Mains 2020 Hard
- If \(f: \mathbf{N} \rightarrow \mathbf{Z}\) is defined by
\(f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix}\), \(k \in \mathbf{N}\),
and \(\sum_{n=1}^{k} f(n) = 98\), then \(k\) is equal to :JEE Mains 2026 Hard - Let \(P \left( a _1, b _1\right)\) and \(Q \left( a _2, b _2\right)\) be two distinct points on a circle with center \(C (\sqrt{2}, \sqrt{3})\). Let \(O\) be the origin and \(OC\) be perpendicular to both \(CP\) and \(CQ\). If the area of the triangle \(OCP\) is \(\frac{\sqrt{35}}{2}\), then \(a _1^2+ a _2^2+ b _1^2+ b _2^2\) is equal to \(...........\).JEE Mains 2023 Hard
- Let \(f :[-3,1] \rightarrow R\) be given as \(f(x)=\left\{\begin{array}{ll} \min \left\{(x+6), x^{2}\right\}, & -3 \leq x \leq 0 \\ \max \left\{\sqrt{x}, x^{2}\right\}, & 0 \leq x \leq 1 \end{array}\right.\) If the area bounded by \(y = f ( x )\) and \(x\) -axis is \(A,\) then the value of \(6 A\) is equal to ....... .JEE Mains 2021 Hard
- If \(C_{x} \equiv^{25} C_{x}\) and \(\mathrm{C}_{0}+5 \cdot \mathrm{C}_{1}+9 \cdot \mathrm{C}_{2}+\ldots .+(101) \cdot \mathrm{C}_{25}=2^{25} \cdot \mathrm{k}\) then \(\mathrm{k}\) is equal toJEE Mains 2020 Hard
- Let \(\hat{a}\) be a unit vector perpendicular to the vectors \(\overrightarrow{\mathrm{b}}=\hat{i}-2 \hat{j}+3 \hat{k}\) and \(\overrightarrow{\mathrm{c}}=2 \hat{i}+3 \hat{j}-\hat{k}\), and makes an angle of \(\cos ^{-1}\left(-\frac{1}{3}\right)\) with the vector \(\hat{i}+\hat{j}+\hat{k}\). If \(\hat{\mathrm{a}}\) makes an angle of \(\frac{\pi}{3}\) with the vector \(\hat{i}+\alpha \hat{j}+\hat{k}\), then the value of \(\alpha\) is :JEE Mains 2025 Medium
More PYQs from JEE Mains
- The value of \({\cos ^2}\,{10^o}\,\, - \,\cos \,\,{10^o}\,\cos \,\,{50^o}\, + \,{\cos ^2}\,{50^o}\) isJEE Mains 2019 Hard
- If a circle passing through the point \((-1, 0)\) touches \(y-\) axis at \((0, 2)\), then the length of the chord of the circle along the \(x-\) axis isJEE Mains 2015 Hard
- Let \(v\) be the solution of the differential equation \(\left(1-x^{2}\right) d y=\left(xy+\left(x^{3}+2\right) \sqrt{1-x^{2}}\right) d x,-1 < x < 1\) and \(y(0)=0\) if \(\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) d x=k\) then \(k^{-1}\) is equal to:JEE Mains 2022 Hard
- If \(\int\left( e ^{2 x }+2 e ^{ x }- e ^{- x }-1\right) e ^{\left( e ^{ x }+ e ^{- x }\right)} d x\) \(=g(x) e^{\left(e^{x}+e^{-x}\right)}+c,\) where \(c\) is a constant of integration, then \(g (0)\) is equal toJEE Mains 2020 Hard
- Let \(C\) be the set of all complex numbers. Let \(\mathrm{S}_{1}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-2| \leq 1\} \text { and }\) \(\mathrm{S}_{2}=\{\mathrm{z} \in \mathrm{C}: \mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i}) \geq 4\}\) Then, the maximum value of \(\left|z-\frac{5}{2}\right|^{2}\) for \(z \in \mathrm{S}_{1} \cap \mathrm{S}_{2}\) is equal to:JEE Mains 2021 Hard
- If \(\vec a,\vec b\) and \(\vec c\) are unit vectors such that \(\vec a + 2\vec b + 2\vec c = \vec 0\), then \(\left| {\vec a \times \vec c} \right|\) is equal toJEE Mains 2018 Hard