1
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1

If $$\int\limits_0^\pi {{{{5^{\cos x}}(1 + \cos x\cos 3x + {{\cos }^2}x + {{\cos }^3}x\cos 3x)dx} \over {1 + {5^{\cos x}}}} = {{k\pi } \over {16}}}$$, then k is equal to _____________.

2
JEE Main 2023 (Online) 1st February Morning Shift
Numerical
+4
-1

If $$\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}$$ where $$l, m, n \in \mathbb{N}, m$$ and $$n$$ are coprime then $$l+m+n$$ is equal to ____________.

3
JEE Main 2023 (Online) 1st February Morning Shift
Numerical
+4
-1

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a differentiable function such that $$f^{\prime}(x)+f(x)=\int_\limits{0}^{2} f(t) d t$$. If $$f(0)=e^{-2}$$, then $$2 f(0)-f(2)$$ is equal to ____________.

4
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1

$$\lim_\limits{x \rightarrow 0} \frac{48}{x^{4}} \int_\limits{0}^{x} \frac{t^{3}}{t^{6}+1} \mathrm{~d} t$$ is equal to ___________.