List of top Mathematics Questions on Methods of Integration asked in JEE Advanced

For any y∈R, let cot$^{-1}$(y) ∈ (0,π) and tan$^{-1}$(y) ∈ ($-\frac{\pi}{2},\frac{\pi}{2}$). Then the sum of all the solutions of the equation$tan^{-1}(\frac{6y}{9-y^2})+cot^{-1}(\frac{9-y^2}{6y})=\frac{2\pi}{3}\, for\,0<|y|<3,$ is equal to

Let f : (0,1) → R be the function defined as f(x) = √n if x ∈ [$\frac{1}{n+1},\frac{1}{n}$] where n ∈ N. Let g : (0,1) → R be a function such that $\int_{x^2}^{x}\sqrt{\frac{1-t}{t}}dt<g(x)<2\sqrt x$ for all x ∈ (0,1).
Then $\lim_{x\rightarrow0}f(x)g(x)$

For $x ∈ R$, let $tan^{-1} (x) ∈ $ ($-\frac{\pi}{2},\frac{\pi}{2}$). Then the minimum value of function $f: R \rightarrow R$ defined by $f(x) =$ $\int_{0}^{x\,tan^{-1}x}\frac{e^{(t-cos\,t)}}{1+t^{2023}}dt$ is

Let f:[1,∞) $\rightarrow$ R be a differentiable function such that f(1) = $\frac{1}{3}$ and 3$\int_{1}^{x}$f(t)dt=xf(x)-$\frac{x^3}{3}$,x∈[1,∞). Let e denote the base of the natural logarithm. Then the value of f(e) is

Let n ≥ 2 be a natural number and f:[0,1)]$\rightarrow$ R be the function defined by
$f(x) \begin{cases} n(1-2nx)\ \ \ \ \ \ \ \ \ \text{if}\ 0\le x\le\frac{1}{2n} \\ 2n((2nx-1)\ \ \ \ \ \text{if}\ \frac{1}{2n}\le x\le \frac{3}{4n}\\4n(1-nx)\ \ \ \ \ \ \ \ \ \text{if}\ \frac{3}{4n}\le x\le\frac{1}{n} \\ \frac{n}{n-1}(nx-1)\ \ \ \ \ \ \ \text{if}\ \frac{1}{n}\le x\le1 \end{cases}$
If n is such that the area of the region bounded by the curves x = 0, x = 1, y = 0 and y = f(x) is 4, then the maximum value of the function f is

Let n ≥ 2 be a natural number and f:[0,1)]$\rightarrow$ R be the function defined by

If n is such that the area of the region bounded by the curves x = 0, x = 1, y = 0 and y = f(x) is 4, then the maximum value of the function f is

Let S be the set of all twice differentiable functions f from R to R such that $\frac{d^2f}{dx^2}(x)>0$ for all x∈(-1,1). For f∈S, let X_f be the number of points x∈(-1,1) for which f(x)=x.Then which of the following statements is(are) true?

Which of the following inequalities is/are TRUE?

Let the function $f:[0,1] \rightarrow R$ be defined by
$f(x)=\frac{4^{x}}{4^{x}+2}$.
Then the value of
$f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\ldots+f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$
is ____