Indefinite Integration PYQs Set 5 (PDF)

Set 5 of 16 · Questions 121-150 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 5 (Q121-150)
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121. $When x > 0, then \int cos^{- 1} (\frac{1 - x ^{2}}{1 + x ^{2}}) dx is$ (1) $2 (x tan^{- 1} x - log (1 + x^{2})) + c$ (2) $2 (x tan^{- 1} x + log (1 + x^{2})) + c$ (3) $2 x tan^{- 1} x + log (1 + x^{2}) + c$ (4) $2 x tan^{- 1} x - log (1 + x^{2}) + c$ 122. $\int \frac{d x}{s i n ( x - \frac{π}{3} ) c o s x} =$ (1) $2 log (\frac{tan x - 3}{2}) + C$ (2) $2 log (sin (x - \frac{π}{3}) cos x) + C$ (3) $2 log (tan x + 3) + C$ (4) $2 log (sin x + 3 cos x) + C$ 123. $\int_{5}^{9} \frac{2}{log 3 x ^{2} + log ( 588 - 84 x + 3 x ^{2} )} dx is equal to$ (1) $2$ (2) $1$ (3) $\frac{1}{2}$ (4) $4$ 124. $The value of \int e^{x} (\frac{1 + sin x}{1 + cos x}) dx is equal to$ (1) $e^{x} tan \frac{x}{2} + C$ (2) $e^{x} tan x + C$ (3) $e^{x} (1 + cos x) + C$ (4) $e^{x} (1 + sin x) + C$ 125. $Find integration \int \frac{dx}{( x - 3 ) ^{6/7} \cdot ( x + 4 ) ^{8/7}}$ (1) $(\frac{x - 3}{x + 4})^{\frac{1}{7}} + c$ (2) $7 (\frac{x - 3}{x + 4})^{\frac{1}{7}} + c$ (3) $7 (\frac{x - 3}{x + 4})^{\frac{6}{7}} + c$ (4) $7 (\frac{x + 4}{x - 3})^{\frac{6}{7}} + c$ 126. $Consider the following statements (A) and (B)(A) \int_{a}^{b} \frac{d}{dx} (f (x)) dx = \frac{d}{dx} \int_{a}^{b} (f (x)) dx (B) \frac{d}{dx} (\int f (x) dx) = f (x) + C Which one of the following is true?$ (1) $Only (A) is true$ (2) $Only (B) is true$ (3) $Both (A) and (B) are true$ (4) $Both (A) and (B) are false$ 127. $\int cose c x - si n x dx =$ (1) $2 sin x + C$ (2) $sin x + C$ (3) $\frac{2}{sin x} + C$ (4) $\frac{sin x}{2} + C$ 128. $The integral \int sec^{2/3} x cosec^{4/3} xdx is equal to:$ (1) $- 3 tan^{- 1/3} x + C$ (2) $- \frac{3}{4} tan^{- 4/3} x + C$ (3) $- 3 cot^{- 1/3} x + C$ (4) $3 tan^{- 1/3} x + C$ 129. $\int \frac{dx}{( 1 + x ) ^{2022}} =$ (1) $\frac{2}{( 1 + x ) ^{2021}} (\frac{1 + x}{2020} - \frac{1}{2021}) + C$ (2) $\frac{2}{( 1 + x ) ^{2022}} (\frac{1 + x}{2020} - \frac{x}{2021}) + C$ (3) $\frac{2}{( 1 + x )} (\frac{( 1 + x ) ^{2022}}{2022} - \frac{( 1 + x ) ^{2021}}{2021}) + C$ (4) $\frac{1}{( 1 + x ) ^{2}} (\frac{1}{( 1 + x ) ^{1010}} - \frac{1}{( 1 + x ) ^{1011}}) + C$ 130. $If \frac{42 - 13 x}{x ^{2} + x - 6} = \frac{A}{lx + m} + \frac{B}{px + q} where lm > 0 and pq < 0 then \frac{Alp}{Bmq} =$ (1) $\frac{27}{32}$ (2) $\frac{27}{8}$ (3) $\frac{8}{243}$ (4) $\frac{243}{32}$ 131. $\int_{- 3}^{3} cot^{- 1} xdx =$ (1) $3 π$ (2) $6 π$ (3) $0$ (4) $3$ 132. $\int \frac{x ^{3}}{1 + x ^{2}} dx is equal to$ (1) $1 + x^{2} - \frac{x}{3} (1 + x^{2})^{3/2} + C$ (2) $x 1 + x^{2} + \frac{2}{3} (1 + x^{2})^{3/2} + C$ (3) $x^{2} 1 + x^{2} - \frac{2}{3} (1 + x^{2})^{3/2} + C$ (4) $x^{2} 1 + x^{2} - \frac{1}{3} (1 + x^{2})^{1/2} + C$ 133. $If [x] is the greatest integer function not greater than x then \int_{0}^{8} [x] dx is equal to$ (1) $28$ (2) $29$ (3) $30$ (4) $20$ 134. $If \int \frac{cos θ}{5 + 7 sin θ - 2 cos ^{2} θ} dθ = A log_{e} \| f (θ) \| + c (where c is a constant of integration), then \frac{f ( θ )}{A} can be$ (1) $\frac{2 sin θ + 1}{sin θ + 3}$ (2) $\frac{2 sin θ + 1}{5 ( sin θ + 3 )}$ (3) $\frac{5 ( sin θ + 3 )}{2 sin θ + 1}$ (4) $\frac{5 ( 2 sin θ + 1 )}{sin θ + 3}$ 135. $lim n \to \infty (\frac{1}{1 + n ^{5}} + \frac{2 ^{4}}{2 ^{5} + n ^{5}} + \frac{3 ^{4}}{3 ^{5} + n ^{5}} + \dots + \frac{n ^{4}}{n ^{5} + n ^{5}}) =$ (1) $\frac{1}{5} log 3$ (2) $\frac{1}{3} log 5$ (3) $\frac{1}{2} log 5$ (4) $log 52$ 136. $\int_{0}^{π} \frac{cos ^{4} x}{cos ^{4} x + sin ^{4} x} dx =$ (1) $\frac{π}{4}$ (2) $\frac{π}{2}$ (3) $\frac{π}{8}$ (4) $π$ 137. $If f (x) = \int_{x}^{x + 1} e^{- t^{2}} dt, then the interval in which f (x) is decreasing is$ (1) $(- \frac{1}{2}, \infty)$ (2) $(- \infty, 2)$ (3) $(- \infty, 0)$ (4) $(- 2, 2)$ 138. $If \frac{x ^{3}}{( 2 x - 1 ) ( x + 2 ) ( x - 3 )} = A + \frac{B}{2 x - 1} + \frac{C}{x + 2} + \frac{D}{x - 3} then A is equal to$ (1) $\frac{1}{2}$ (2) $\frac{- 1}{50}$ (3) $\frac{- 8}{25}$ (4) $\frac{27}{25}$ 139. $If I_{n} = \int_{0}^{π /4} tan^{n} xdx, then \frac{1}{I _{2} + I _{4}} + \frac{1}{I _{3} + I _{5}} + \frac{1}{I _{4} + I _{6}} =$ (1) $\frac{1}{I _{9} + I _{11}}$ (2) $\frac{1}{I _{10} + I _{12}}$ (3) $\frac{1}{I _{12} + I _{14}}$ (4) $\frac{1}{I _{11} + I _{13}}$ 140. $For x^{2} \neq = nπ + 1, n \in N (the set of natural numbers), the integral \int x \frac{2 sin ( x ^{2} - 1 ) - sin 2 ( x ^{2} - 1 )}{2 sin ( x ^{2} - 1 ) + sin 2 ( x ^{2} - 1 )} dx is equal to:$ (1) $log_{e} (\frac{1}{2} sec^{2} (x^{2} - 1)) + c$ (2) $\frac{1}{2} log_{e} (sec (x^{2} - 1)) + c$ (3) $\frac{1}{2} log_{e} (sec^{2} (\frac{x ^{2} - 1}{2})) + c$ (4) $log_{e} (sec (\frac{x ^{2} - 1}{2})) + c$ 141. $Evaluate: \int \frac{x ^{3} + x}{x ^{4} - 9} dx$ (1) $\frac{1}{4} log (x^{4} - 9) + \frac{1}{12} log (\frac{x ^{2} + 3}{x ^{2} - 3}) + C$ (2) $\frac{1}{4} log (x^{4} - 9) - \frac{1}{12} log (\frac{x ^{2} - 3}{x ^{2} + 3}) + C$ (3) $\frac{1}{4} log (x^{4} - 9) + \frac{1}{12} log (\frac{x ^{2} - 3}{x ^{2} + 3}) + C$ (4) $None of these$ 142. $If the equivalent partial fraction of is of the form then the value of$ (1) $- 8/25$ (2) $4/25$ (3) $- 1/50$ (4) $1/2$ 143. $If \int e^{- x} tan^{- 1} (e^{x}) dx = f (x) - \frac{1}{2} log (1 + e^{2 x}) + C, then f (x) equals$ (1) $e^{x} - e^{- x} tan^{- 1} (e^{x})$ (2) $x^{2} + e^{- x} tan^{- 1} (e^{x})$ (3) $- e^{- x} tan^{- 1} (e^{x})$ (4) $x - e^{- x} tan^{- 1} (e^{x})$ 144. $If \frac{x - 2}{x ^{2} ( 2 x - 3 )} = \frac{A}{x} + \frac{B}{x ^{2}} + \frac{C}{2 x - 3} then 2 (A - C) =$ (1) $3 B$ (2) $2 B$ (3) $0$ (4) $B$ 145. $If \int \frac{dx}{( x + 2 ) ( x ^{2} + 1 )} = a log (1 + x^{2}) + b tan^{- 1} x + \frac{1}{5} log \| x + 2 \| + c, then$ (1) $a = \frac{- 1}{10}, b = \frac{2}{5}$ (2) $a = \frac{- 1}{10}, b = \frac{- 2}{5}$ (3) $a = \frac{1}{10}, b = \frac{2}{5}$ (4) $a = \frac{1}{10}, b = \frac{- 2}{5}$ 146. $\int_{0}^{π} \frac{x tan x}{sec x \cdot cosec x} dx =$ (1) $\frac{π}{2}$ (2) $\frac{π}{4}$ (3) $\frac{π ^{2}}{2}$ (4) $\frac{π ^{2}}{4}$ 147. $If x > 0 and x \neq = (2 n + 1) \frac{π}{2} then \int (x x - e^{log (sec xtan x)} + \frac{3 x ^{2} - 2 x + 1}{x ^{2}}) dx =$ (1) $x x - sec x + 3 x - 2 lo g x - \frac{1}{x} + c$ (2) $\frac{2}{5} x^{2} x - sec x + 3 x + \frac{2}{x ^{2}} - \frac{1}{x} + c$ (3) $x x - sec x + 3 x + \frac{2}{x ^{2}} - \frac{1}{x} + c$ (4) $\frac{2}{5} x^{2} x - sec x + 3 x - 2 lo g x - \frac{1}{x} + c$ 148. $\int \frac{( 1 - cos x ) ^{2/7}}{( 1 + cos x ) ^{9/7}} dx is equal to$ (1) $\frac{7}{11} (tan \frac{x}{2})^{\frac{11}{7}} + C$ (2) $\frac{7}{11} (tan \frac{x}{2})^{\frac{7}{11}} + C$ (3) $\frac{7}{11} (cot \frac{x}{2})^{\frac{11}{7}} + C$ (4) $\frac{11}{7} (cot \frac{x}{2})^{\frac{7}{11}} + C$ 149. $If \frac{3 x ^{2} + ax + 3}{( 2 x + 3 ) ( x ^{2} + 2 )} = \frac{3}{2 x + 3} + \frac{Bx + C}{x ^{2} + 2} then a (B + C) =$ (1) $-2$ (2) $3$ (3) $-3$ (4) $2$ 150. $If \int \frac{2 x ^{2} + 5 x + 9}{x ^{2} + x + 1} d x = x x^{2} + x + 1 + α x^{2} + x + 1 + β lo g_{e} (x + \frac{1}{2} + x^{2} + x + 1) + C, where C is the constant of integration, then α + 2 β is equal to .$