Indefinite Integration PYQs Set 1 (PDF)

Set 1 of 16 · Questions 1-30 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 1 (Q1-30)
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1. $If \int \frac{1 - x ^{4}}{x ^{7}} dx = f (x) (1 - x^{4})^{n} + C, then (f (x))^{n} is equal to$ (1) $\frac{- 1}{6 x ^{6}}$ (2) $\frac{- 1}{216 x ^{18}}$ (3) $\frac{1}{36 x ^{12}}$ (4) $\frac{1}{216 x ^{18}}$ 2. $\sum \frac{cos 7 x - cos 8 x}{1 + 2 cos 5 x} dx =$ (1) $sin 2 x - \frac{1}{3} sin 3 x + c$ (2) $\frac{1}{2} sin 2 x - \frac{1}{3} sin 3 x + c$ (3) $\frac{1}{2} sin 2 x - sin 3 x + c$ (4) $\frac{1}{3} sin 2 x - \frac{1}{2} sin 3 x + c$ 3. $If f (x) = \int (tan^{2} x + cot^{2} x + \frac{4 ( sin ^{3} x + cos ^{3} x )}{sin ^{2} 2 x}) dx and f (\frac{π}{4}) = 0, then 3 (f (\frac{π}{6}) + 2) =$ (1) $\frac{π}{2}$ (2) $\frac{π}{4}$ (3) $0$ (4) $\frac{- π}{2}$ 4. $If x \neq = 1 and \sum \frac{( x ^{3} + x ^{2} - x - 1 )}{( x ^{5} + x ^{4} + 3 x ^{3} + 3 x ^{2} + x + 1 ) tan ^{- 1} ( \frac{x ^{2} + 1}{x} )} d x = Alog (f (x)) + C, then A - tan (f (2)) =$ (1) $\frac{- 3}{2}$ (2) $\frac{- 1}{2}$ (3) $\frac{7}{2}$ (4) $-2$ 5. $\sum 4 cos^{2} x - 5 sin^{2} x cos xdx =$ (1) $\frac{1}{2} sin x 4 - 9 sin^{2} x + \frac{2}{3} sin^{- 1} (\frac{3 sin x}{2}) + c$ (2) $\frac{1}{2} cos x 4 - 9 cos^{2} x + \frac{2}{3} sin^{- 1} (\frac{3 cos x}{2}) + c$ (3) $\frac{1}{2} sin x 4 - 9 sin^{2} x + \frac{2}{3} cos^{- 1} (\frac{3 cos x}{2}) + c$ (4) $\frac{1}{2} cos x 4 - 9 sin^{2} x + \frac{2}{3} sin^{- 1} (\frac{3 sin x}{2}) + c$ 6. $If \int \frac{dx}{cos ^{4} x + sin ^{4} x} = \frac{1}{2} tan^{- 1} [g (x)] + C, then g (x) equals$ (1) $\frac{tan x - cot x}{2}$ (2) $\frac{tan x + cot x}{2}$ (3) $\frac{sin x - cos x}{2}$ (4) $\frac{sin x + cos x}{2}$ 7. $\int_{1}^{2} \frac{x ^{3} - 1}{x ^{2}} dx is equal to$ (1) $\frac{5}{3}$ (2) $\frac{3}{5}$ (3) $1$ (4) $-1$ 8. $If \int \frac{sin ^{2} α - sin ^{2} x}{cos x - cos α} dx = f (x) + Ax + B and B \in R, then$ (1) $f (x) = 2 sin x, A = cos α$ (2) $f (x) = 2 sin x, A = 2 cos α$ (3) $f (x) = sin x, A = cos α$ (4) $f (x) = sin x, A = 2 cos α$ 9. $\int \frac{cos ^{n - 1} x}{sin ^{n + 1} x} dx, n \neq = 0 is$ (1) $\frac{c o t ^{n} x}{n} + C$ (2) $- \frac{c o t ^{n - 1} x}{n - 1} + C$ (3) $- \frac{c o t ^{n} x}{n} + C$ (4) $\frac{c o t ^{n - 1} x}{n - 1} + C$ 10. $The value of \int e^{sin x} sin 2 xdx is$ (1) $2 e^{sin x} (sin x + 1) + C$ (2) $2 e^{sin x} (cos x + 1) + C$ (3) $2 e^{sin x} (cos x - 1) + C$ (4) $2 e^{sin x} (sin x - 1) + C$ 11. $Let f (x) = \int \frac{e ^{3 x}}{4 + 8 e ^{2 x} + e ^{4 x}} dx and g (x) = \int \frac{2 dx}{e ^{3 x} + 8 e ^{x} + 4 e ^{- x}} dx, then f (x) - g (x) =$ (1) $\frac{1}{2} tan^{- 1} (\frac{e ^{x} + 2 e ^{- x}}{2}) + C$ (2) $\frac{1}{2} tan^{- 1} (\frac{e ^{x} + e ^{- x}}{2}) + C$ (3) $\frac{1}{2} tan^{- 1} (\frac{2 e ^{- x} + e ^{- 2 x}}{2}) + C$ (4) $\frac{1}{2} tan^{- 1} (\frac{e ^{2 x} + 2 e ^{x}}{2 e ^{- x}}) + C$ 12. $\int \frac{s e c ^{2} x}{( s e c x + t a n x ) ^{2}} d x =$ (1) $\frac{3 + ( sec x + tan x ) ^{2}}{2 ( sec x + tan x ) ^{3}} + c$ (2) $- \frac{1 + 3 ( sec x + tan x ) ^{2}}{6 ( sec x + tan x ) ^{3}} + c$ (3) $- \frac{3 + ( sec x + tan x ) ^{2}}{2 ( sec x + tan x ) ^{3}} + c$ (4) $- \frac{1 + ( sec x + tan x )}{3 ( sec x + tan x ) ^{2}} + c$ 13. $\int 3^{- log_{9} x^{2}} dx =$ (1) $2 log \| x \| + C$ (2) $log \| x \| + C$ (3) $- log \| x \| + C$ (4) $- 2 log \| x \| + C$ 14. $The value of \int_{0}^{1} \frac{log ( 1 + x )}{1 + x ^{2}} dx is$ (1) $\frac{π}{4} log 2$ (2) $\frac{1}{2}$ (3) $\frac{π}{8} log 2$ (4) $\frac{π}{2} log 2$ 15. $lim n \to \infty \frac{1}{n} (\frac{1}{e ^{1/ n}} + \frac{1}{e ^{2/ n}} + \frac{1}{e ^{3/ n}} + \dots + \frac{1}{e ^{2}}) =$ (1) $1 - e^{- 2}$ (2) $1 + e^{- 2}$ (3) $e^{2} - 1$ (4) $e^{2} + 1$ 16. $If \int \frac{x + 1}{2 x - 1} d x = f (x) 2 x - 1 + C where C is a constant of integration, then f (x) is equal to:$ (1) $\frac{1}{3} (x + 1)$ (2) $\frac{2}{3} (x + 2)$ (3) $\frac{2}{3} (x - 4)$ (4) $\frac{1}{3} (x + 4)$ 17. $It is given that \frac{d}{dt} (tlog t - t) = log t then exp (\int_{0}^{1} 2 xlog (1 + x^{2}) dx) =$ (1) $e$ (2) $2$ (3) $\frac{4}{e}$ (4) $\frac{e}{4}$ 18. $\int_{1}^{4} (x + x + \frac{1}{x}) dx - \int_{1}^{2 log 2} dx =$ (1) $\frac{79}{6}$ (2) $\frac{643}{6}$ (3) $\frac{321}{5}$ (4) $64$ 19. $The integral \int cos (log_{e} x) dx is equal to : (where C is a constant of integration)$ (1) $\frac{x}{2} (sin (log_{e} x) - cos (log_{e} x)) + C$ (2) $x (cos (log_{e} x) + sin (log_{e} x)) + C$ (3) $\frac{x}{2} (cos (log_{e} x) + sin (log_{e} x)) + C$ (4) $x (cos (log_{e} x) - sin (log_{e} x)) + C$ 20. $If \int \frac{dθ}{cos ^{2} θ ( tan 2 θ + sec 2 θ )} = λ tan θ + 2 log_{e} \| f (θ) \| + C where c is a constant of integration,then the ordered pair (λ, f (θ)) is equal to:$ (1) $(1, 1 + tan θ)$ (2) $(- 1, 1 - tan θ)$ (3) $(- 1, 1 + tan θ)$ (4) $(1, 1 - tan θ)$ 21. $\frac{x ^{4}}{( x ^{2} + 1 ) ( x ^{2} + 3 )} =$ (1) $\frac{Ax + B}{x ^{2} + 1} + \frac{Cx + D}{x ^{2} + 3} for some A, B, C, D \in R ∖{0}$ (2) $\frac{Ax + B}{x ^{2} + 1} + \frac{Cx}{x ^{2} + 1} for some A, B, C \in R ∖{0}$ (3) $\frac{Ax}{x ^{2} + 1} + \frac{Bx}{x ^{2} + 3} for some A,B \in R ∖{0}$ (4) $1 + \frac{Ax + B}{x ^{2} + 1} + \frac{Cx + D}{x ^{2} + 3} for some A, B, C, D \in R$ 22. $\int_{- 1}^{1} \frac{1 + x + x ^{2} - 1 - x + x ^{2}}{1 + x + x ^{2} + 1 - x + x ^{2}} dx is equal to$ (1) $\frac{3 π}{2}$ (2) $\frac{π}{2}$ (3) $0$ (4) $-1$ 23. $\sum \frac{e ^{cot x}}{sin ^{2} x} (2 log cosec x + sin 2 x) dx =$ (1) $- 2 e^{cot x} log (cosec^{2} x) + C$ (2) $- 2 e^{cot x} log (cosec x) + C$ (3) $- 2 e^{cot x} log (cosec x + sin x) + C$ (4) $- 2 e^{cot x} log (cosec x - cot x) + C$ 24. $If \int \frac{dx}{x ( x ^{4} - 1 )} = \frac{1}{k} sec^{- 1} (x^{k}), then the value of k is equal to$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 25. $\int_{- 2}^{2} \| x cos πx \| dx is equal to$ (1) $\frac{8}{π}$ (2) $\frac{4}{π}$ (3) $\frac{2}{π}$ (4) $\frac{1}{π}$ 26. $\int e^{tan^{- 1} x} (1 + \frac{x}{1 + x ^{2}}) dx is equal to$ (1) $\frac{1}{2} e^{tan^{- 1} x} + c$ (2) $\frac{1}{2} x e^{tan^{- 1} x} + c$ (3) $x e^{tan^{- 1} x} + c$ (4) $e^{tan^{- 1} x} + c$ 27. $The integra \int \frac{3 x ^{13} + 2 x ^{11}}{( 2 x ^{4} + 3 x ^{2} + 1 ) ^{4}} dx is equal to: (where C is a constant of integration)$ (1) $\frac{x ^{4}}{6 ( 2 x ^{4} + 3 x ^{2} + 1 ) ^{3}} + C$ (2) $\frac{x ^{12}}{6 ( 2 x ^{4} + 3 x ^{2} + 1 ) ^{3}} + C$ (3) $\frac{x ^{4}}{( 2 x ^{4} + 3 x ^{2} + 1 ) ^{3}} + C$ (4) $\frac{x ^{12}}{( 2 x ^{4} + 3 x ^{2} + 1 ) ^{3}} + C$ 28. $If \sum \frac{sin ^{\frac{3}{2}} x + cos ^{\frac{3}{2}} x}{sin ^{3} x cos ^{3} x sin ( x - θ )} dx = A cos θ tan x - sin θ + B cos θ - sin θ cot x + C where C is the integration constant, then AB is equal to$ (1) $4 cosec (2 θ)$ (2) $4 sec θ$ (3) $2 sec θ$ (4) $8 cosec (2 θ)$ 29. $Let f (x) = \int \frac{2 x ^{3} - 3 x ^{2} + 4 x - 5}{x ^{2}} dx and f (1) = 1 . Then f (5) =$ (1) $10 + 4 log 5$ (2) $10 - 4 log 5$ (3) $9 + 4 log 5$ (4) $9 - 4 log 5$ 30. $The value of \int_{0}^{4042} \frac{x dx}{x + 4042 - x} is equal to$ (1) $4042$ (2) $2021$ (3) $8084$ (4) $1010$