Definite Integration PYQs Set 5 (PDF)

Set 5 of 9 · Questions 121-150 of 248

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Definite Integration · Set 5 (Q121-150)
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121. $If φ (x) = \frac{1}{x} \int_{π /4}^{x} (42 sin t - 3 φ^{'} (t)) d t, x > 0 then φ^{'} (\frac{π}{4}) is equal to$ (1) $\frac{8}{. π}$ (2) $\frac{4}{6 + π}$ (3) $\frac{8}{6 + π}$ (4) $\frac{4}{6 - π}$ 122. $The value of \int_{- π /6}^{π /6} \frac{π + 4 x ^{11}}{1 - s i n ( ∣ x ∣ + π /6 )} d x is equal to:$ (1) $8 π$ (2) $6 π$ (3) $2 π$ (4) $4 π$ 123. $Let f : [1, \infty) \to [2, \infty) be a differentiable function. If 10 \int_{1}^{x} f (t) d t = 5 x f (x) - x^{5} - 9 for all x \geq 1, then the value of f (3) is:$ (1) $22$ (2) $26$ (3) $32$ (4) $18$ 124. $If I_{n} = \int_{π /4}^{π /2} cot^{n} x d x, then$ (1) $\frac{1}{I _{2} + I _{4}}, \frac{1}{I _{3} + I _{5}}, \frac{1}{I _{4} + I _{6}} are in G.P$ (2) $\frac{1}{I _{2} + I _{4}}, \frac{1}{I _{3} + I _{5}}, \frac{1}{I _{4} + I _{6}} are in A.P$ (3) $I_{2} + I_{4}, I_{3} + I_{5}, I_{4} + I_{6} are in A.P$ (4) $I_{2} + I_{4}, (I_{3} + I_{5})^{2}, I_{4} + I_{6} are in G.P$ 125. $For x > 0, if f (x) = \int_{1}^{x} \frac{l o g _{e} t}{1 + t} d t then f (e) + f (\frac{1}{e}) is equal to$ (1) $\frac{1}{2}$ (2) $- 1$ (3) $1$ (4) $0$ 126. $\int_{0}^{3} [x] d x =____, where [x] is greatest integer function.$ (1) $3$ (2) $0$ (3) $2$ (4) $1$ 127. $The value of the integral \int_{1/2}^{2} \frac{t a n ^{- 1} x}{x} d x is equal to$ (1) $π log_{e} 2$ (2) $\frac{1}{2} log_{2} 2$ (3) $\frac{π}{4} log_{e} 2$ (4) $\frac{π}{2} log_{e} 2$ 128. $The integral \int_{0}^{π} \frac{8 x d x}{4 c o s ^{2} x + s i n ^{2} x} is equal to$ (1) $2 π^{2}$ (2) $4 π^{2}$ (3) $π^{2}$ (4) $\frac{3 π ^{2}}{2}$ 129. $The value of \int_{- 1}^{1} x^{2} e^{[x^{3}]} d x where [t] denotes the greatest integer \leq t is$ (1) $\frac{e + 1}{3}$ (2) $\frac{e - 1}{3 e}$ (3) $\frac{e + 1}{3 e}$ (4) $\frac{1}{3 e}$ 130. $The value of \int_{π /3}^{π /2} \frac{2 + 3 s i n x}{s i n x ( 1 + c o s x )} d x is equal to$ (1) $\frac{7}{2} - 3 - lo g_{e} 3$ (2) $- 2 + 33 + lo g_{e} 3$ (3) $\frac{10}{3} - 3 + lo g_{e} 3$ (4) $\frac{10}{3} - 3 - lo g_{e} 3$ 131. $If \int_{- π /2}^{π /2} \frac{96 x ^{2} c o s ^{2} x}{1 + e ^{x}} d x = π (α π^{2} + β), α, β \in Z, then (α + β)^{2} equals$ (1) $64$ (2) $196$ (3) $144$ (4) $100$ 132. $Let f be a polynomial function such that f (x^{2} + 1) = x^{4} + 5 x^{2} + 2, for all x \in R . Then \int_{0}^{3} f (x) d x is equal to$ (1) $\frac{5}{3}$ (2) $\frac{41}{3}$ (3) $\frac{27}{2}$ (4) $\frac{33}{2}$ 133. $The integral \int_{- 1}^{3/2} π^{2} x sin (π x) d x is equal to:$ (1) $2 + 3 π$ (2) $4 + π$ (3) $1 + 3 π$ (4) $3 + 2 π$ 134. $If the value of the integral \int_{0}^{1/2} \frac{x ^{2}}{( 1 - x ^{2} ) ^{3/2}} d x = \frac{k}{6}, then k is equal to$ (1) $23 + π$ (2) $32 - π$ (3) $32 + π$ (4) $23 - π$ 135. $If f (a + b + 1 - x) = f (x), for all x, where a and b are fixed positive real numbers, then \frac{1}{( a + b )} \int_{a}^{b} x (f (x) + f (x + 1)) d x is equal to$ (1) $\int_{a + 1}^{b + 1} f (x) d x$ (2) $\int_{a + 1}^{b + 1} f (x + 1) d x$ (3) $\int_{a - 1}^{b - 1} f (x) d x$ (4) $\int_{a - 1}^{b - 1} f (x + 1) d x$ 136. $Let f (x) = \int_{0}^{x} g (t) d t, where g is non zero even function. If f (x + 5) = g (x), then \int_{0}^{x} g (t) d t equals:$ (1) $\int_{x + 5}^{5} g (t) d t$ (2) $\int_{5}^{x + 5} g (t) d t$ (3) $2 \int_{5}^{x + 5} g (t) d t$ (4) $5 \int_{x + 5}^{5} g (t) d t$ 137. $If \int_{0}^{π /3} \frac{t a n θ}{2 k s e c θ} d θ = 1 - \frac{1}{2}, (k > 0) then the value of k is :$ (1) $4$ (2) $\frac{1}{2}$ (3) $1$ (4) $2$ 138. $The value of the integral \int_{π /24}^{5 π /24} \frac{d x}{1 + 3 t a n 2 x} is :$ (1) $\frac{π}{3}$ (2) $\frac{π}{12}$ (3) $\frac{π}{18}$ (4) $\frac{π}{6}$ 139. $The value of \int_{- π /2}^{π /2} \frac{c o s ^{2} x}{1 + 3 ^{x}} d x is$ (1) $2 π$ (2) $4 π$ (3) $\frac{π}{2}$ (4) $\frac{π}{4}$ 140. $The integral \int_{0}^{π} \frac{( x + 3 ) s i n x}{1 + 3 c o s ^{2} x} d x is equal to$ (1) $\frac{π}{3} (π + 1)$ (2) $\frac{π}{3 3} (π + 6)$ (3) $\frac{π}{3} (π + 2)$ (4) $\frac{π}{2 3} (π + 4)$ 141. $The value of the integral \int_{1}^{2} \frac{t ^{4} + 1}{t ^{6} + 1} d t is$ (1) $tan^{- 1} \frac{1}{2} + \frac{1}{3} tan^{- 1} 8 - \frac{π}{3}$ (2) $tan^{- 1} 2 - \frac{1}{3} tan^{- 1} 8 + \frac{π}{3}$ (3) $tan^{- 1} 2 + \frac{1}{3} tan^{- 1} 8 - \frac{π}{3}$ (4) $tan^{- 1} \frac{1}{2} - \frac{1}{3} tan^{- 1} 8 + \frac{π}{3}$ 142. $Let I_{n} = \int_{1}^{e} x^{19} (lo g ∣ x ∣)^{n} d x,where n \in N . If (20) I_{10} = α I_{9} + β I_{8}, for natural numbers α and β, then α - β equal to ____.$ 143. $Let f : (0, \infty) \to R be a twice differentiable function. If for some a \neq = 0, \int_{0}^{1} f (λ x) d λ = a f (x), f (1) = 1 and f (16) = \frac{1}{8}, then 16 - f^{'} (\frac{1}{16}) is equal to .$ 144. $If \int_{0}^{1} (x^{21} + x^{14} + x^{7}) (2 x^{14} + 3 x^{7} + 6)^{1/7} d x = \frac{1}{l} (11)^{m / n} where l, m, n \in N, m and n are coprime then l + m + n is equal to _______$ 145. $Let S = (- 1, \infty) and f : S \to R be defined as f (x) = \int_{- 1}^{x} (e^{t} - 1)^{11} (2 t - 1)^{5} (t - 2)^{7} (t - 3)^{12} (2 t - 10)^{61} d t . Let p = Sum of square of the values of x, where f (x) attains local maxima on S and q = Sum of the values of x, where f (x) attains local minima on S . Then, the value of p^{2} + 2 q is ________$ 146. $The value of \sum_{r = 1}^{20} (π (\int_{0}^{r} x ∣ sin π x ∣ d x)) is ____$ 147. $If [\cdot] represents the greatest integer function,then the value of \int_{0}^{π /2} [[x^{2}] - cos x] d x is _____$ 148. $The value 9 \int_{0}^{9} [\frac{10 x}{x + 1}] d x, where [t] denotes the greatest integer less than or equal to t, is$ 149. $The number of elements in the set S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} sin (x - t) d t = x^{2}} is ____.$ 150. $If the integral \int_{0}^{10} \frac{[ s i n 2 π x ]}{e ^{x - [x]}} d x = α e^{- 1} + β e^{- \frac{1}{2}} + γ, where α, β, γ are integers and [x] denotes the greatest integer less than or equal to x, then the value of α + β + γ is equal to : (a) 0 (b) 20 (c) 25 (d) 10$