Definite Integration: JEE Main Mathematics Previous Year Questions

248 previous year questions from Definite Integration. Here are the 25 most important ones; open any question for the correct answer and a full step-by-step solution.

1. 1.Let $g(x)={\int }_0^{x}f(t)dt$, where $f$ is continuous function in [0,3] such that $\frac{1}{3}\le f(t)\le 1$ for all $t\in (0,1)$ and $0\le f(t)\le \frac{1}{2}$ for all…
2. 2.If ${\int }_0^K\frac{\text{dx}}{2+18x^2}=\frac{\pi }{24}$, then the value of $K$ is
3. 3.${\int }_0^4\frac{1}{1+\sqrt{x}}\text{dx}=$
4. 4.Let a differentiable function $f$ satisfy $f(x)+{\int }_3^x\frac{f(t)}{t}dt=\sqrt{x+1},x\ge 3$. Then $12f(8)$ is equal to
5. 5.The value of the integral ${\int }_{-1}^2{\text{log}}_e\text{⁡}(x+\sqrt{x^2+1})\text{dx}$ is
6. 6.…
7. 7.A value of $\alpha$ such that ${\int }_{\alpha }^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}={\log }_e\left(\frac{9}{8}\right)$ is:
8. 8.The integral ${\int }_1^e\left({\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x\right){\log }_{e}xdx$ is equal to :
9. 9.Let$f\text{:}\left(0\text{,}2\right)\to R$ be defined as $f\left(x\right)={\text{log}}_2(1+\text{tan}\left(\frac{\text{πx}}{4}\right))$ Then,…
10. 10.The value of the integral ${\int }_{-2}^2\frac{{\sin }^{2}x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}dx$ (where [x] denotes the greatest integer less than or equal to x) is:
11. 11.If ${\int }_0^1\frac{1}{\sqrt{3+x}+\sqrt{1+x}}\text{dx}=a+b\sqrt{2}+c\sqrt{3}$, where $a\text{,}b\text{,}c$ are rational numbers, then…
12. 12.If the value of the integral…
13. 13.Let $f\text{:}R\to R$ be defined as $f\left(x\right)=e^{-x}\text{sin}x$. If $F\text{:}\left(0\text{,}1\right)\to R$ is a differentiable function such that…
14. 14.The value of ${\int }_{-\pi }^{\pi }\frac{2y\text{(}1+\text{sin}\text{⁡}y\text{)}}{1+{\text{cos}}^2\text{⁡}y}\text{dy}$ is:
15. 15.The integral ${\int }_1^2{e}^x\cdot x^x(2+{\text{log}}_e\text{⁡}x)\text{dx}$ equals:
16. 16.…
17. 17.The value of ${\int }_0^{2\pi }\left[\sin 2x(1+\cos 3x)\right]dx$, where [t] denotes the greatest integer function, is:
18. 18.Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of ${\int }_1^2|2x-[3x]|dx$ is
19. 19.The value of $12{\int }_0^3|x^2-3x+2|dx$ is ________
20. 20.Let $r_k=\frac{{\int }_0^1{(1-x^7)}^k\text{dx}}{{\int }_0^1{(1-x^7)}^{k+1}\text{dx}}\text{,}k\text{∈}N$…
21. 21.${\int }_0^{1}x{\text{tan}}^{-1}\text{⁡}\text{xdx}=$
22. 22.The value of ${\text{lim}}_{n\to \infty }{\sum }_{k=1}^n\frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is:
23. 23.${\int }_0^{1}x\text{(}1-x{\text{)}}^5\text{dx}=$
24. 24.If $\sum \left(\frac{1-5{\text{cos}}^{2}x}{{\text{sin}}^{5}x{\text{cos}}^{2}x}\right)\text{dx}=f\text{(}x\text{)}+C$, where C is the constant of integratio…
25. 25.Let $f\text{:}R\to R$ be a twice differentiable function such that $f\left(2\right)=1$. If $F\left(x\right)=\text{xf}\left(x\right)$ for all $x$ $R$,…