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JEE Main

Explore popular questions from Integral Calculus for JEE Main. This collection covers Integral Calculus previous year JEE Main questions hand picked by experienced teachers.

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Integral Calculus

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Q 1. Area bounded by lines {tex} y = 2 + x , y = 2 - x {/tex} and {tex} x = 2 {/tex} is

A

3

4

C

8

D

16

Explanation


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Q 2. The area bounded by the curves {tex} y = \log _ { e } x {/tex} and {tex} y = \left( \log _ { e } x \right) ^ { 2 } {/tex} is

{tex} 3 - e {/tex}

B

{tex} e - 3 {/tex}

C

{tex} \frac { 1 } { 2 } ( 3 - e ) {/tex}

D

{tex} \frac { 1 } { 2 } ( e - 3 ) {/tex}

Explanation


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Q 3. The area enclosed by the parabolas {tex} y = x ^ { 2 } - 1 {/tex} and {tex} y = 1 - x ^ { 2 } {/tex} is

A

1{tex} / 3 {/tex}

B

2{tex} / 3 {/tex}

C

4{tex} / 3 {/tex}

8{tex} / 3 {/tex}

Explanation


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Q 4. The area enclosed between the parabolas {tex} y ^ { 2 } = 4 x {/tex} and {tex} x ^ { 2 } = 4 y {/tex} is

A

{tex} \frac { 14 } { 3 } {/tex} sq. units

B

{tex} \frac { 3 } { 4 } {/tex} sq.units

C

{tex} \frac { 3 } { 16 } {/tex} sq. units

{tex} \frac { 16 } { 3 } {/tex} sq. units

Explanation


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Q 5. The area bounded by the curves {tex} y ^ { 2 } = 8 x {/tex} and {tex} y = x {/tex} is

A

{tex} \frac { 128 } { 3 } {/tex} sq. units

{tex} \frac { 32 } { 3 } {/tex} sq. units

C

{tex} \frac { 64 } { 3 } {/tex} sq. units

D

{tex}32{/tex} sq.units

Explanation



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Q 6. The area bounded by the parabola {tex} y ^ { 2 } = 4 a x , {/tex} its axis and two ordinates {tex} x = 4 , x = 9 {/tex} is

A

4{tex} a ^ { 2 } {/tex}

B

4{tex} a ^ { 2 } \cdot 4 {/tex}

C

4{tex} a ^ { 2 } ( 9 - 4 ) {/tex}

{tex} \frac { 152 \sqrt { a } } { 3 } {/tex}

Explanation


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Q 7. The area of the region consisting of points {tex} ( x , y ) {/tex} satisfying {tex} | x \pm y | \leq 2 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } \geq 2 {/tex} is

{tex} 8 - 2 \pi {/tex} sq.units

B

{tex} 4 - 2 \pi {/tex} sq.units

C

{tex} 1 - 2 \pi {/tex} sq.units

D

{tex} 2 \pi \ {/tex} sq.units

Explanation


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Q 8. The area of the curve {tex} x y ^ { 2 } = a ^ { 2 } ( a - x ) {/tex} bounded by {tex} y {/tex} -axis is

{tex} \pi a ^ { 2 } {/tex}

B

2{tex} \pi a ^ { 2 } {/tex}

C

3{tex} \pi a ^ { 2 } {/tex}

D

4{tex} \pi a ^ { 2 } {/tex}

Explanation


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Q 9. The area of the region bounded by the curves {tex} y = x ^ { 2 } {/tex} and {tex} y = | x | {/tex} is

A

1{tex} / 6 {/tex}

1{tex} / 3 {/tex}

C

{tex}5 / 6 {/tex}

D

{tex} 5/ 3 {/tex}

Explanation


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Q 10. Let {tex} g ( x ) {/tex} be a function defined on {tex} [ - 1,1 ] {/tex} . If the area of the equilateral triangle with two of its vertices at {tex} ( 0,0 ) {/tex} and {tex} [ x , g ( x ) ] {/tex}
is {tex} \frac { \sqrt { 3 } } { 4 } , {/tex} then the function {tex} g ( x ) {/tex} is

{tex} g ( x ) = \pm \sqrt { 1 - x ^ { 2 } } {/tex}

B

{tex} g ( x ) = \sqrt { 1 - x ^ { 2 } } {/tex}

C

{tex} g ( x ) = - \sqrt { 1 - x ^ { 2 } } {/tex}

D

{tex} g ( x ) = \sqrt { 1 + x ^ { 2 } } {/tex}

Explanation



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Q 11. The area (in square units) of the region bounded by the curves {tex} y + 2 x ^ { 2 } = 0 {/tex} and {tex} y + 3 x ^ { 2 } = 1 {/tex} is equal to

A

{tex} \frac { 3 } { 5 } {/tex}

B

{tex} \frac { 3 } { 4 } {/tex}

C

{tex} \frac { 1 } { 3 } {/tex}

{tex} \frac { 4 } { 3 } {/tex}

Explanation


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Q 12. If a curve {tex} y = a \sqrt { x } + b x {/tex} passes through the point {tex} ( 1,2 ) {/tex} and the area bounded by the curve, line {tex} x = 4 {/tex} and {tex} x {/tex} -axis is 8 sq. units, then

{tex}a=3 , b=-1{/tex}

B

{tex} a = 3 , b = 1 {/tex}

C

{tex} a = - 3 , b = 1 {/tex}

D

{tex} a = - 3 , b = - 1 {/tex}

Explanation



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Q 13. Area under the curve {tex} y = \sin 2 x + \cos 2 x {/tex} between {tex} x = 0 {/tex} and {tex} x = \frac { \pi } { 4 } {/tex} is

A

{tex} 2 {/tex} sq.units

{tex} 1 {/tex} sq.units

C

{tex}3{/tex} sq.units

D

{tex} 4 {/tex} sq.units

Explanation


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Q 14. The area between the parabola {tex} y ^ { 2 } = 4 a x {/tex} and {tex} x ^ { 2 } = 8 a y {/tex} is

A

{tex} \frac { 8 } { 3 } a ^ { 2 } {/tex}

B

{tex} \frac { 4 } { 3 } a ^ { 2 } {/tex}

{tex} \frac { 32 } { 3 } a ^ { 2 } {/tex}

D

{tex} \frac { 16 } { 3 } a ^ { 2 } {/tex}

Explanation


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Q 15. For {tex} 0 \leq x \leq \pi , {/tex} the area bounded by {tex} y = x {/tex} and {tex} y = x + \sin x {/tex} is

2

B

4

C

2{tex} \pi {/tex}

D

4{tex} \pi {/tex}

Explanation


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Q 16. The area of the region bounded by the {tex} x {/tex} -axis and the curves defined by {tex} y = \tan x , ( - \pi / 3 \leq x \leq \pi / 3 ) {/tex} is

A

{tex} \log \sqrt { 2 } {/tex}

B

{tex} - \log \sqrt { 2 } {/tex}

2 {tex} \log 2 {/tex}

D

{tex} 0 {/tex}

Explanation


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Q 17. The area between the curve {tex} y ^ { 2 } = 4 a x , x {/tex}-axis and the ordinates {tex} x = 0 {/tex} and {tex} x = a {/tex} is

A

{tex} \frac { 4 } { 3 } a ^ { 2 } {/tex}

{tex} \frac { 8 } { 3 } a ^ { 2 } {/tex}

C

{tex} \frac { 2 } { 3 } a ^ { 2 } {/tex}

D

{tex} \frac { 5 } { 3 } a ^ { 2 } {/tex}

Explanation



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Q 18. The value of {tex} \sqrt { 2 } \int \frac { \sin x d x } { \sin \left( x - \frac { \pi } { 4 } \right) } {/tex} is

A

{tex} x + \log \left| \cos \left( x - \frac { \pi } { 4 } \right) \right| + c {/tex}

B

{tex} x - \log \left| \sin \left( x - \frac { \pi } { 4 } \right) \right| + c {/tex}

{tex} x + \log \left| \sin \left( x - \frac { \pi } { 4 } \right) \right| + c {/tex}

D

{tex} x - \log \left| \cos \left( x - \frac { \pi } { 4 } \right) \right| + c {/tex}

Explanation


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Q 19. The integral {tex} \int \left( 1 + x - \frac { 1 } { x } \right) e ^ { x + \frac { 1 } { x } } d x {/tex} is equal to

A

{tex} ( x + 1 ) e ^ { x + \frac { 1 } { x } } + c {/tex}

B

{tex} - x e ^ { x + \frac { 1 } { x } } + c {/tex}

C

{tex} ( x - 1 ) e ^ { x + \frac { 1 } { x } } + c {/tex}

{tex} x e ^ { x + \frac { 1 } { x } } + c {/tex}

Explanation


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Q 20. {tex} \int \frac { \sin ^ { 8 } x - \cos ^ { 8 } x } { \left( 1 - 2 \sin ^ { 2 } x \cos ^ { 2 } x \right) } d x {/tex} is equal to

A

{tex} \frac { 1 } { 2 } \sin 2 x + c {/tex}

{tex} - \frac { 1 } { 2 } \sin 2 x + c {/tex}

C

{tex} - \frac { 1 } { 2 } \sin x + c {/tex}

D

{tex} - \sin ^ { 2 } x + c {/tex}

Explanation


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Q 21. The integral {tex} \int \frac { \sin ^ { 2 } x \cos ^ { 2 } x } { \left( \sin ^ { 3 } x + \cos ^ { 3 } x \right) ^ { 2 } } d x {/tex} equal to

A

{tex} \frac { 1 } { \left( 1 + \cot ^ { 3 } x \right) } + c {/tex}

{tex} - \frac { 1 } { 3 \left( 1 + \tan ^ { 3 } x \right) } + c {/tex}

C

{tex} \frac { \sin ^ { 3 } x } { \left( 1 + \cos ^ { 3 } x \right) } + c {/tex}

D

{tex} - \frac { \cos ^ { 3 } x } { 3 \left( 1 + \sin ^ { 3 } x \right) } + c {/tex}

Explanation


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Q 22. If {tex} m {/tex} is a non-zero number and {tex} \int \frac { x ^ { 5 m - 1 } + 2 x ^ { 4 m - 1 } } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 3 } } d x = f ( x ) + c {/tex} then {tex} f ( x ) {/tex} is

A

{tex} \frac { x ^ { 5 m } } { 2 m \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } } {/tex}

{tex} \frac { x ^ { 4 m } } { 2 m \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } } {/tex}

C

{tex} \frac { 2 m \left( x ^ { 5 m } + x ^ { 4 m } \right) } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } } {/tex}

D

{tex} \frac { \left( x ^ { 5 m } - x ^ { 4 m } \right) } { 2 m \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } } {/tex}

Explanation


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Q 23. The integral {tex} \int \frac { d x } { ( 1 + \sqrt { x } ) \sqrt { x - x ^ { 2 } } } {/tex} is equal to (where {tex} C {/tex} is a constant of integration)

A

{tex} - 2 \sqrt { \frac { 1 + \sqrt { x } } { 1 - \sqrt { x } } } + c {/tex}

B

{tex} - \sqrt { \frac { 1 - \sqrt { x } } { 1 + \sqrt { x } } } + c {/tex}

{tex} - 2 \sqrt { \frac { 1 - \sqrt { x } } { 1 + \sqrt { x } } } + c {/tex}

D

{tex} 2 \sqrt { \frac { 1 + \sqrt { x } } { 1 - \sqrt { x } } } + c {/tex}

Explanation


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Q 24. The value of {tex} \int \frac { 1 + x + x ^ { 2 } } { 1 + x ^ { 2 } } e ^ { \tan ^ { - 1 } x } d x {/tex} is equal to

A

{tex} x ^ { 2 } e ^ { \tan ^ { - 1 } x } {/tex}

B

{tex} e ^ { \tan ^ { - 1 } x } + c {/tex}

{tex} x e ^ { \tan ^ { - 1 } x } + c {/tex}

D

None of these

Explanation


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Q 25. {tex} \int \frac { \tan x } { \sqrt { \cos x } } d x {/tex} is equal to

A

{tex} \frac { 2 } { \sqrt { \sin x } } + c {/tex}

{tex} \frac { 2 } { \sqrt { \cos x } } + c {/tex}

C

{tex} \frac { 2 } { \sqrt { \tan x } } + c {/tex}

D

{tex} \frac { 2 } { ( \sin x ) ^ { 3 / 2 } } + c {/tex}

Explanation