Definite Integration PYQs Set 1 (PDF)

Set 1 of 9 · Questions 1-30 of 248

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Definite Integration · Set 1 (Q1-30)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
1. $Let g (x) = \int_{0}^{x} f (t) d t, where f is continuous function in [0,3] such that \frac{1}{3} \leq f (t) \leq 1 for all t \in (0, 1) and 0 \leq f (t) \leq \frac{1}{2} for all t \in (1, 3) . The largest possible interval in which g (3) lies:$ (1) $(- 1, - \frac{1}{2})$ (2) $(- \frac{3}{2}, - 1)$ (3) $(\frac{1}{3}, 2)$ (4) $(1, 3)$ 2. $If \int_{0}^{K} \frac{dx}{2 + 18 x ^{2}} = \frac{π}{24}, then the value of K is$ (1) $3$ (2) $4$ (3) $\frac{1}{3}$ (4) $\frac{1}{4}$ 3. $\int_{0}^{4} \frac{1}{1 + x} dx =$ (1) $log (\frac{e ^{4}}{6})$ (2) $log (\frac{e ^{4}}{3})$ (3) $log (\frac{e ^{4}}{9})$ (4) $log (\frac{e ^{3}}{4})$ 4. $Let a differentiable function f satisfy f (x) + \int_{3}^{x} \frac{f ( t )}{t} d t = x + 1, x \geq 3 . Then 12 f (8) is equal to$ (1) $34$ (2) $19$ (3) $17$ (4) $1$ 5. $The value of the integral \int_{- 1}^{2} log_{e} (x + x^{2} + 1) dx is$ (1) $5 - 2 + log_{e} (\frac{7 + 4 5}{1 + 2})$ (2) $5 - 2 + log_{e} (\frac{9 + 4 5}{1 + 2})$ (3) $2 - 5 + log_{e} (\frac{7 + 4 5}{1 + 2})$ (4) $2 - 5 + log_{e} (\frac{9 + 4 5}{1 + 2})$ 6. $lim n \to \infty (\frac{( n + 1 ) ^{\frac{1}{3}}}{n ^{\frac{4}{3}}} + \frac{( n + 2 ) ^{\frac{1}{3}}}{n ^{\frac{4}{3}}} + \dots + \frac{( 2 n ) ^{\frac{1}{3}}}{n ^{\frac{4}{3}}}) is equal to :$ (1) $\frac{3}{4} (2)^{\frac{4}{3}} - \frac{3}{4}$ (2) $\frac{4}{3} (2)^{\frac{4}{3}}$ (3) $\frac{3}{2} (2)^{\frac{4}{3}} - \frac{4}{3}$ (4) $\frac{4}{3} (2)^{\frac{3}{4}}$ 7. $A value of α such that \int_{α}^{α + 1} \frac{d x}{( x + α ) ( x + α + 1 )} = lo g_{e} (\frac{9}{8}) is:$ (1) $-2$ (2) $\frac{1}{2}$ (3) $- \frac{1}{2}$ (4) $2$ 8. $The integral \int_{1}^{e} ((\frac{x}{e})^{2 x} - (\frac{e}{x})^{x}) lo g_{e} x d x is equal to :$ (1) $\frac{1}{2} - e - \frac{1}{e ^{2}}$ (2) $- \frac{1}{2} + \frac{1}{e} - \frac{1}{2 e ^{2}}$ (3) $\frac{3}{2} - \frac{1}{e} - \frac{1}{2 e ^{2}}$ (4) $\frac{3}{2} - e - \frac{1}{2 e ^{2}}$ 9. $The value of the integral \int_{- 2}^{2} \frac{s i n ^{2} x}{[ \frac{x}{π} ] + \frac{1}{2}} d x (where [x] denotes the greatest integer less than or equal to x) is:$ (1) $0$ (2) $sin 4$ (3) $4$ (4) $4 - sin 4$ 10. $If \int_{0}^{1} \frac{1}{3 + x + 1 + x} dx = a + b 2 + c 3, where a, b, c are rational numbers, then 2 a + 3 b - 4 c is equal to :$ (1) $4$ (2) $10$ (3) $7$ (4) $8$ 11. $If the value of the integral \sum (\frac{x ^{2} cos x}{1 + π ^{x}} + \frac{1 + sin ^{2} x}{1 + e ^{sin x^{2033}}}) dx = \frac{π}{4} (π + a) - 2, then the value of a is$ (1) $3$ (2) $- \frac{3}{2}$ (3) $2$ (4) $\frac{3}{2}$ 12. $Let f : R \to R be defined as f (x) = e^{- x} sin x . If F : (0, 1) \to R is a differentiable function such that F (x) = \sum f (t) dt, then the value of \sum (F^{’} (x) + f (x)) e^{x} dx lies in the interval$ (1) $(\frac{327}{360}, \frac{329}{360})$ (2) $(\frac{330}{360}, \frac{331}{360})$ (3) $(\frac{331}{360}, \frac{334}{360})$ (4) $(\frac{335}{360}, \frac{336}{360})$ 13. $The value of \int_{- π}^{π} \frac{2 y ( 1 + sin y )}{1 + cos ^{2} y} dy is:$ (1) $2 π^{2}$ (2) $\frac{π ^{2}}{2}$ (3) $\frac{π}{2}$ (4) $π^{2}$ 14. $The integral \int_{1}^{2} e^{x} \cdot x^{x} (2 + log_{e} x) dx equals:$ (1) $e (2 e - 1)$ (2) $e (4 e - 1)$ (3) $4 e^{2} - 1$ (4) $e (4 e + 1)$ 15. $lim x \to \infty (\frac{1}{n} + \frac{n}{( n + 1 ) ^{2}} + \frac{n}{( n + 2 ) ^{2}} + \dots \dots + \frac{n}{( 2 n - 1 ) ^{2}}) is equal to$ (1) $1$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $\frac{1}{4}$ 16. $The value of \int_{0}^{2 π} [sin 2 x (1 + cos 3 x)] d x, where [t] denotes the greatest integer function, is:$ (1) $π$ (2) $- π$ (3) $- 2 π$ (4) $2 π$ 17. $\int_{0}^{1} x tan^{- 1} xdx =$ (1) $\frac{π}{4} + \frac{1}{2}$ (2) $\frac{π}{4} - \frac{1}{2}$ (3) $\frac{1}{2} - \frac{π}{4}$ (4) $- \frac{π}{4} - \frac{1}{2}$ 18. $The value of lim_{n \to \infty} \sum_{k = 1}^{n} \frac{n ^{3}}{( n ^{2} + k ^{2} ) ( n ^{2} + 3 k ^{2} )} is:$ (1) $\frac{( 2 3 + 3 ) π}{24}$ (2) $\frac{13 π}{8 ( 4 3 + 3 )}$ (3) $\frac{13 ( 2 3 - 3 ) π}{8}$ (4) $\frac{mπ}{8 ( 2 3 + 3 )}$ 19. $\int_{0}^{1} x (1 - x)^{5} dx =$ (1) $1/5$ (2) $1/42$ (3) $1/13$ (4) $13/42$ 20. $If \sum (\frac{1 - 5 cos ^{2} x}{sin ^{5} x cos ^{2} x}) dx = f (x) + C, where C is the constant of integration, then f (\frac{π}{6}) - f (\frac{π}{4}) is equal to$ (1) $\frac{1}{3} (26 + 3)$ (2) $\frac{1}{3} (26 - 3)$ (3) $\frac{4}{3} (8 - 6)$ (4) $\frac{2}{3} (4 + 6)$ 21. $Let f : R \to R be a twice differentiable function such that f (2) = 1 . If F (x) = xf (x) for all x R, \sum x F^{’} (x) d x = 6 and \sum x^{2} F^{’ ’} (x) d x = 40, then F^{’} (2) + \sum F (x) d x is equal to:$ (1) $11$ (2) $13$ (3) $15$ (4) $9$ 22. $The value of \int_{- 1}^{1} \frac{( 1 + \| x \| - x ) e ^{x} + ( \| x \| - x ) e ^{- x}}{e ^{x} + e ^{- x}} dx is equal to$ (1) $1 + \frac{2 2}{3}$ (2) $1 - \frac{2 2}{3}$ (3) $2 + \frac{2 2}{3}$ (4) $3 - \frac{2 2}{3}$ 23. $Find the value of \int_{0}^{2 π} \frac{x sin ^{8} x}{sin ^{8} x + cos ^{8} x} dx$ (1) $π^{2}$ (2) $2 π^{2}$ (3) $3 π^{2}$ (4) $4 π^{2}$ 24. $The value of \int_{- π /2}^{π /2} \frac{d x}{[ x ] + [ s i n x ] + 4}, where [t] denotes the greatest integer less than or equal to t, is:$ (1) $\frac{1}{12} (7 π + 5)$ (2) $\frac{1}{12} (7 π - 5)$ (3) $\frac{3}{20} (4 π - 3)$ (4) $\frac{3}{10} (4 π - 3)$ 25. $The value of \int_{- π /2}^{π /2} (cos x)^{3} d x is :$ (1) $0$ (2) $\frac{4}{3}$ (3) $\frac{2}{3}$ (4) $- \frac{4}{3}$ 26. $The value of \int_{- π /2}^{π /2} \frac{1}{1 + e ^{sin x}} dx is :$ (1) $\frac{π}{4}$ (2) $π$ (3) $\frac{3 π}{2}$ (4) $\frac{π}{2}$ 27. $Let f : (0, 2) \to R be defined as f (x) = log_{2} (1 + tan (\frac{πx}{4})) Then, lim n \to \infty \frac{2}{n} (f (\frac{1}{n}) + f (\frac{2}{n}) + \dots . + f (1)) is equal to ______$ 28. $Let [t] denote the greatest integer less than or equal to t . Then the value of \int_{1}^{2} ∣2 x - [3 x] ∣ d x is$ 29. $The value of 12 \int_{0}^{3} ∣ x^{2} - 3 x + 2∣ d x is ________$ 30. $Let r_{k} = \frac{\int _{0}^{1} ( 1 - x ^{7} ) ^{k} dx}{\int _{0}^{1} ( 1 - x ^{7} ) ^{k + 1} dx}, k \in N . Then the value of \sum_{k = 1}^{10} \frac{1}{7 ( r _{k} - 1 )} is equal to________$