JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of the integral \(\int\limits_{-1}^{1} \left(\dfrac{x^3 + |x| + 1}{x^2 + 2|x| + 1}\right) dx\) is equal to :
- A \(3\log_e 2\)
- B \(2\log_e 2\)
- C \(5\log_e 3\)
- D \(3\log_e 3\)
Answer & Solution
Correct Answer
(B) \(2\log_e 2\)
Step-by-step Solution
Detailed explanation
Let \(I = \int_{-1}^{1} \left(\dfrac{x^3 + |x| + 1}{x^2 + 2|x| + 1}\right) dx\) We can split the integral into two parts: \(I = \int_{-1}^{1} \dfrac{x^3}{x^2 + 2|x| + 1} dx + \int_{-1}^{1} \dfrac{|x| + 1}{x^2 + 2|x| + 1} dx\) The first integrand…
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
- Let \(A =\{0,1,2, \ldots 9)\). Let R be a relation on A defined by \(( x , y ) \in R\) if and only if \(| x - y |\) is a multiple of 3 .
Given below are two statements:
Statement I: \(n ( R )=36\)
Statement II: \(R\) is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given belowJEE Mains 2026 Hard - Let the values of p , for which the shortest distance between the lines \(\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}\) and \(\overrightarrow{\mathrm{r}}=(\mathrm{p} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})\) is \(\frac{1}{\sqrt{6}}\), be \(\mathrm{a}, \mathrm{b}\), \((a \lt b)\). Then the length of the latus rectum of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is :-JEE Mains 2025 Medium
- Let \(P \left( x _0, y _0\right)\) be the point on the hyperbola \(3 x ^2-4 y ^2\) \(=36\), which is nearest to the line \(3 x+2 y=1\). Then \(\sqrt{2}\left( y _0- x _0\right)\) is equal to :JEE Mains 2023 Hard
- The area of the triangle with vertices \(A ( z ), B ( iz )\) and \(C(z+i z)\) isJEE Mains 2021 Medium
- If \(\lim _{\mathrm{t} \rightarrow 0}\left(\int_0^1(3 x+5)^{\mathrm{t}} \mathrm{d} x\right)^{\frac{1}{t}}=\frac{\alpha}{5 \mathrm{e}}\left(\frac{8}{5}\right)^{\frac{2}{3}}\), then \(\alpha\) is equal to ________JEE Mains 2025 Hard
- If the range of \(f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}\) is \([\alpha, \beta]\), then the sum of the infinite \(G.P.\), whose first term is \(64\) and the common ratio is \(\frac{\alpha}{\beta}\), is equal to ...........JEE Mains 2024 Hard
More PYQs from JEE Mains
- The shortest distance between the lines \(\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3}\) and \(\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5}\) isJEE Mains 2023 Easy
- Let the centre of a circle, passing through the point \((0,0),(1,0)\) and touching the circle \(x^2+y^2=9\), be \((h, k)\). Then for all possible values of the coordinates of the centre \((h, k), 4\left(h^2+k^2\right)\) is equal to .............JEE Mains 2024 Hard
- If the curve \(y = f(x)\) passes through the point \((1, e)\) and satisfies the differential equation \(dy = y(2 + \log_e x)\,dx\), \(x > 0\), then \(f(e)\) is equal to :JEE Mains 2026 Medium
- Given a sequence of \(4\) numbers, first three of which are in \(G.P.\) and the last three are in \(A.P\). with common difference six. If first and last terms in this sequence are equal, then the last term isJEE Mains 2013 Hard
- If the probability that the random variable X takes the value \(x\) is given by \(P(X=x)=k(x+1) 3^{-x}\), \(\mathrm{x}=0,1,2,3 \ldots \ldots\), where k is a constant, then \(\mathrm{P}(\mathrm{X} \geq 3)\) is equal toJEE Mains 2025 Medium
- If \(f(x)\) is a quadratic expression such that \(f(1) + f (2)\, = 0\) , and \(-1\) is a root of \(f(x)\, = 0\), then the other root of \(f(x)\, = 0\) isJEE Mains 2018 Hard