JEE Mains · Maths · STD 12 - 7.1 indefinite integral
Let \( f(x)=\int\frac{7x^{10}+9x^{8}}{(1+x^{2}+2x^{9})^{2}}dx \), \( x > 0 \), \( \lim_{x\rightarrow0}f(x)=0 \) and \( f(1)=\frac{1}{4} \). If \( A=\begin{bmatrix}0&0&1\\ \frac{1}{4}&f'(1)&1\\ \alpha^{2}&4&1\end{bmatrix} \) and \( B=adj(adj~A) \) be such that \( |B|=81 \), then \( \alpha^{2} \) is equal to
- A 2
- B 3
- C 1
- D 4
Answer & Solution
Correct Answer
(D) 4
Step-by-step Solution
Detailed explanation
\( f(x)=\int\frac{(\frac{7}{x^{8}}+\frac{9}{x^{10}})}{(\frac{1}{x^{9}}+\frac{1}{x^{7}}+2)^{2}}dx \) Put \( t=\frac{1}{x^{9}}+\frac{1}{x^{7}}+2 \Rightarrow \frac{dt}{dx}=\frac{-9}{x^{10}}-\frac{7}{x^{8}} \) \( f(x)=\int\frac{-dt}{t^{2}}=\frac{1}{t}+C \)…
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 for positive integers \(r> 1, n > 2\), the coefficients of the \((3r)^{th}\) and \((r + 2)^{th}\) powers of \(x\) in the expansion of \(( 1 + x)^{2n}\) are equal, then \(n\) is equal toJEE Mains 2013 Hard
- Consider the following frequency distribution :
If mean \(=\frac{309}{22}\) and median \(=14\), than value \((a-b)^{2}\) is equal to \(.....\)Class: \(0-6\) \(6-12\) \(12-18\) \(18-24\) \(24-30\) Frequency : \(a\) \(b\) \(12\) \(9\) \(5\) JEE Mains 2021 Hard - Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}|=1,|\vec{b}|=4\) and \(\vec{a} \cdot \vec{b}=2\). If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\alpha\), then \(192 \sin ^2 \alpha\) is equal toJEE Mains 2024 Medium
- If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+c\) where \(c\) is a constant of integration, then \(\lambda f\left(\frac{\pi}{3}\right)\) is equal toJEE Mains 2020 Hard
- Let a tangent to the curve \(y^2=24 x\) meet the curve \(xy =2\) at the points \(A\) and \(B\). Then the mid points of such line segments \(A B\) lie on a parabola with theJEE Mains 2023 Hard
- If the complex number \(z=2-i\left(2 \tan \frac{5 \pi}{8}\right)\) has modulus \(r\) and argument \(\theta\), then what are \((r, \theta)\) ?JEE Mains 2024 Medium
More PYQs from JEE Mains
- Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its \(11^{\text {th }}\) term is :JEE Mains 2025 Medium
- Let \(A\) be a matrix of order \(2 \times 2\), whose entries are from the set \(\{0,1,2,3,4,5\}\). If the sum of all the entries of \(A\) is a prime number \(p , 2< p <8\), then the number of such matrices \(A\) isJEE Mains 2022 Hard
- Let \(f: R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{ll}\frac{x^{3}}{(1-\cos 2 x)^{2}} \log _{e}\left(\frac{1+2 x e^{-2 x}}{\left(1-x e^{-x}\right)^{2}}\right), & x \neq 0 \\ \,\alpha & , x=0\end{array}\right.\) If \(\mathrm{f}\) is continuous at \(\mathrm{x}=0\), then \(\alpha\) is equal to :JEE Mains 2021 Hard
- A pole stands vertically inside a triangular park \(ABC\). Let the angle of elevation of the top of the pole from each corner of the park be \(\frac{\pi}{3}\). If the radius of the circumcircle ot \(\Delta ABC\) is \(2 ,\) then the height of the pole is equal to :JEE Mains 2021 Medium
- Let \(a_1, a_2, a_3, \ldots\) be in an arithmetic progression of positive terms. Let \(\mathrm{A}_{\mathrm{k}}=\mathrm{a}_1{ }^2-\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2-\mathrm{a}_4{ }^2+\ldots+\mathrm{a}_{2 \mathrm{k}-1}{ }^2-\mathrm{a}_{2 \mathrm{k}}{ }^2\). If \(\mathrm{A}_3=-153, \mathrm{~A}_5=-435\) and \(\mathrm{a}_1{ }^2+\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2=66\), then \(\mathrm{a}_{17}-\mathrm{A}_7\) is equal to ....................JEE Mains 2024 Hard
- 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