Definite Integration PYQs Set 6 (PDF)

Set 6 of 9 · Questions 151-180 of 248

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Definite Integration · Set 6 (Q151-180)
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151. $If \int_{0}^{x} f (t) d t = x^{2} + \int_{x}^{1} t^{2} f (t) d t, then f^{'} (\frac{1}{2}) is:$ (1) $\frac{24}{25}$ (2) $\frac{18}{25}$ (3) $\frac{4}{5}$ (4) $\frac{6}{25}$ 152. $lim_{n \to \infty} \frac{3}{n} (4 + (2 + \frac{1}{n})^{2} + (2 + \frac{2}{n})^{2} + \dots + (3 - \frac{1}{n})^{2}) is equal to$ (1) $12$ (2) $\frac{19}{3}$ (3) $0$ (4) $19$ 153. $The minimum value of the function f (x) = \int_{0}^{2} e^{∣ x - t ∣} d t is$ (1) $2 (e - 1)$ (2) $2 e - 1$ (3) $2$ (4) $e (e - 1)$ 154. $The value of \int_{0}^{π /2} \frac{s i n ^{3} x}{s i n x + c o s x} d x is:$ (1) $\frac{π - 2}{8}$ (2) $\frac{π - 1}{4}$ (3) $\frac{π - 2}{4}$ (4) $\frac{π - 1}{2}$ 155. $The value of \int_{e^{2}}^{e^{4}} \frac{1}{x} (\frac{e ^{((l o g_{e} x)^{2} + 1)^{- 1}}}{e ^{((l o g_{e} x)^{2} + 1)^{- 1}} + e ^{((6 - l o g_{e} x)^{2} + 1)^{- 1}}}) d x is$ (1) $2$ (2) $log_{e} 2$ (3) $1$ (4) $e^{2}$ 156. $If I = \int_{0}^{π /2} \frac{s i n ^{3/2} x}{s i n ^{3/2} x + c o s ^{3/2} x} d x, then \int_{0}^{π /2} \frac{x s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x equals:$ (1) $\frac{π ^{2}}{12}$ (2) $\frac{π ^{2}}{4}$ (3) $\frac{π ^{2}}{16}$ (4) $\frac{π ^{2}}{8}$ 157. $lim_{x \to 0} \frac{\int _{0}^{x^{2}} ( s i n t ) d t}{x ^{3}} is equal to:$ (1) $\frac{2}{3}$ (2) $0$ (3) $\frac{1}{15}$ (4) $\frac{3}{2}$ 158. $Let [\cdot] denote the greatest integer function. Then \int_{- π /2}^{π /2} (\frac{12 ( 3 + [ x ])}{3 + [ s i n x ] + [ c o s x ]}) d x is equal to :$ (1) $13 π + 1$ (2) $12 π + 5$ (3) $11 π + 2$ (4) $15 π + 4$ 159. $Let α > 0 . If \int_{0}^{α} \frac{x}{x + α - x} d x = \frac{16 + 20 2}{15} then α is equal to$ (1) $2$ (2) $4$ (3) $2$ (4) $22$ 160. $The integral \int_{π /6}^{π /4} \frac{d x}{s i n 2 x ( t a n ^{5} x + c o t ^{5} x )} equals :$ (1) $\frac{1}{20} tan^{- 1} (\frac{1}{9 3})$ (2) $\frac{1}{10} (\frac{π}{4} - tan^{- 1} (\frac{1}{9 3}))$ (3) $\frac{π}{40}$ (4) $\frac{1}{5} (\frac{π}{4} - tan^{- 1} (\frac{1}{9 3}))$ 161. $Let f (x) = \int_{0}^{x} e^{t} f (t) d t + e^{x} be a differentiable function for all x \in R . Then f (x) equals.$ (1) $2 e^{(e^{x} - 1)} - 1$ (2) $e^{(e^{x} - 1)}$ (3) $2 e^{e^{x}} - 1$ (4) $e^{e^{x}} - 1$ 162. $If f (x) = \frac{2 - x c o s x}{2 + x c o s x} and g (x) = lo g_{e} x, (x > 0) then the value of the integral \int_{- π /4}^{π /4} g (f (x)) d x is:$ (1) $lo g_{e} 3$ (2) $lo g_{e} e$ (3) $lo g_{e} 2$ (4) $lo g_{e} 1$ 163. $The value of \int_{0}^{100} e^{x - [x]} d x where [x] is the greatest integer \leq x is:$ (1) $100 (e - 1)$ (2) $100 e$ (3) $100 (1 - e)$ (4) $100 (1 + e)$ 164. $Consider the integral I = \int_{0}^{10} \frac{[ x ] e ^{[x]}}{e ^{x - 1}} d x where [x] denotes the greatest integer less than or equal to x . Then the value of I is equal to:$ (1) $9 (e - 1)$ (2) $40e-1$ (3) $45 (e - 1)$ (4) $9 (e + 1)$ 165. $Let (a, b) be the point of intersection of the curve x^{2} = 2 y and the straight line y - 2 x - 6 = 0 in the second quadrant. Then the integral I = \int_{a}^{b} \frac{9 x ^{2}}{1 + 5 ^{x}} d x is equal to:$ (1) $27$ (2) $18$ (3) $24$ (4) $21$ 166. $The value of \int_{- π /2}^{π /2} \frac{1}{[ x ] + 4} d x, where [\cdot] denotes the greatest integer function, is$ (1) $\frac{1}{60} (21 π - 1)$ (2) $\frac{7}{60} (π - 3)$ (3) $\frac{7}{60} (3 π - 1)$ (4) $\frac{1}{60} (π - 7)$ 167. $The integral 80 \int_{0}^{π /4} (\frac{s i n θ + c o s θ}{9 + 16 s i n 2 θ}) d θ is equal to:$ (1) $3 log_{e} 4$ (2) $4 log_{e} 3$ (3) $6 log_{e} 4$ (4) $2 log_{e} 3$ 168. $Let f (x) be a differentiable function defined on [0, 2] such that f^{'} (x) = f^{'} (2 - x) for all x \in (0, 2), f (0) = 1 and f (2) = e^{2} . Then the value of \int_{0}^{2} f (x) d x is:$ (1) $1 + e^{2}$ (2) $1 - e^{2}$ (3) $2 (1 - e^{2})$ (4) $2 (1 + e^{2})$ 169. $The integral 16 \int_{1}^{2} \frac{d x}{x ^{3} ( x ^{2} + 2 ) ^{2}} is equal to$ (1) $\frac{11}{6} + log_{e} 4$ (2) $\frac{11}{12} + log_{e} 4$ (3) $\frac{11}{12} - log_{e} 4$ (4) $\frac{11}{6} - log_{e} 4$ 170. $The value of \frac{8}{π} \int_{0}^{π /2} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x is ________$ 171. $Let [\cdot] denote the greatest integer function. If \int_{0}^{e^{3}} [\frac{1}{e ^{x - 1}}] d x = α - lo g_{e} 2, then α^{3} is equal to_______.$ 172. $Let a differentiable function f satisfy the equation \int_{0}^{36} f (\frac{t x}{36}) d t = 4 α f (x) . If y = f (x) is a standard parabola passing through the points (2, 1) and (- 4, β), then β^{α} is equal to ____ .$ 173. $If 24 \int_{0}^{π /4} (sin (4 x - \frac{π}{12}) + [2 sin x]) d x = 2 π + α, where [•] denotes the greatest integer function, then α is equal to .$ 174. $Let f (x) and g (x) be two functions satisfying f (x^{2}) + f (4 - x) = 4 x^{3} and g (4 - x) + g (x) = 0, then thevalue of \int_{- 4}^{4} f (x)^{2} d x is:$ 175. $Let f : R \to R be a continuous function such that f (x) + f (x + 1) = 2, for all x \in R . If I_{1} = \int_{0}^{8} f (x) d x and I_{2} = \int_{- 1}^{3} f (x) d x, then the value of I_{1} + 2 I_{2} is equal to _____.$ 176. $The value of the integral \int_{0}^{π} ∣ sin 2 x ∣ d x is$ 177. $Let f be a differentiable function defined on [0, \frac{π}{2}] such that f (x) > 0 and f (x) + \int_{0}^{x} f (t) 1 - (lo g_{e} f (t))^{2} d t = e, x \in [0, \frac{π}{2}] Then (6 lo g_{e} f (\frac{π}{6}))^{2} is equal to ______$ 178. $The integral \int_{0}^{2} ∣ ∣ x - 1∣ - x ∣ d x is equal to$ 179. $In I_{m, n} = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x for m, n \geq 1 and \int_{0}^{1} \frac{x ^{m - 1} + x ^{n - 1}}{( 1 + x ) ^{m + n}} d x = α I_{m, n}, α \in R then α equals.$ 180. $If \int_{- a}^{a} (∣ x ∣ + ∣ x - 2∣) d x = 22, (a > 2) and [x] denotes the greatest integer \leq x, then \int_{a}^{- a} (x + [x]) d x is equal to$