\(∫^{\frac{π}{3}} _0\) \(cos^4x\) \(dx\) is equal to a \(\pi + b\sqrt3\), then \(a^2 + b\) is equal to:
- \(\frac{1}{2}\)
- \(\frac{1}{8}\)
- \(\frac{1}{4}\)
- \(1\)
The Correct Option is B
Solution and Explanation
The Correct Option is (B): \(\frac{1}{8}\)
Top Questions on integral
- The value of the integral \( \int_0^1 x^2 \, dx \) is:
Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \), } \[ \int_0^a f(x) \, dx = f(a), \quad f(1) = 1, \quad f(16) = \frac{1}{8}, \quad \text{then } 16 - f^{-1}\left( \frac{1}{16} \right) \text{ is equal to:}\]
- Let $ f(x) $ be a positive function and $I_1 = \int_{-\frac{1}{2}}^1 2x \, f\left(2x(1-2x)\right) dx$ and $I_2 = \int_{-1}^2 f\left(x(1-x)\right) dx.$ Then the value of $\frac{I_2}{I_1}$ is equal to ____
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Questions Asked in JEE Main exam
- Given below are two statements :
Statement I : Compound (X), shown below, dissolves in \( NaHCO_3 \) solution and has two chiral carbon atoms.
Statement II : Compound (Y), shown below, has two carbons with \( sp^3 \) hybridization, one carbon with \( sp^2 \) and one carbon with \( sp \) hybridization.
In the light of the above statements, choose the correct answer from the options given below :- JEE Main - 2026
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- Let \( [\,\cdot\,] \) denote the greatest integer function, and let \[ f(x) = \min\{\sqrt{2}\,x, x^2\}. \] Let \[ S = \{x \in (-2,2) : \text{the function } g(x) = x[x^2] \text{ is discontinuous at } x\}. \] Then \[ \sum_{x \in S} f(x) \text{ equals} \]
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- Consider the following redox reaction taking place in acidic medium: \[ \mathrm{BH_4^- (aq) + ClO_3^- (aq) \rightarrow H_2BO_3^- (aq) + Cl^- (aq)} \] If the Nernst equation for the above balanced reaction is \[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF}\ln Q, \] then the value of \(n\) is ________ (Nearest integer).
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- The number of $3\times2$ matrices $A$, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is $5$, is
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If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \beta\sqrt{5}}{2}, \] where \( \alpha, \beta \in \mathbb{N} \), then the value of \( \alpha + \beta \) is ___________.
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- Trigonometric Identities
Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.