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

## Chemistry

Integral Calculus

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Q 1. Let {tex} f: R \rightarrow R {/tex} be a continuous and bijective function defined such that {tex} f ( a ) = 0 ( a \neq 0 ) . {/tex} The area bounded by {tex} y = f ( x ) , x = a , x = a - t {/tex} is equal to the area bounded by {tex} y = f ( x ) , x = a , x = a + t \forall t \in R , {/tex} then Graph of {tex} y = f ( x ) {/tex} is symmetrical about point

A

{tex} ( 0,0 ) {/tex}

B

{tex} ( 0 , \alpha ) {/tex}

{tex} ( \alpha , 0 ) {/tex}

D

{tex} ( \alpha , \alpha ) {/tex}

##### Explanation

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Q 2. The value of the integral {tex} \int \frac { \cos ^ { 3 } x + \cos ^ { 5 } x } { \sin ^ { 2 } x + \sin ^ { 4 } x } d x {/tex}

A

{tex} \sin x - 6 \tan ^ { - 1 } ( \sin x ) + c {/tex}

B

{tex} \sin x - 2 ( \sin x ) ^ { - 1 } + c {/tex}

{tex} \sin x - 2 ( \sin x ) ^ { - 1 } - 6 \tan ^ { - 1 } ( \sin x ) + c {/tex}

D

{tex} \sin x - 2 ( \sin x ) ^ { - 1 } + 5 \tan ^ { - 1 } ( \sin x ) + c {/tex}

##### Explanation

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Q 3. If {tex} \int _ { \sin x } ^ { 1 } t ^ { 2 } f ( t ) d t = 1 - \sin x , {/tex} then {tex} \mathrm { f } \left( \frac { 1 } { \sqrt { 3 } } \right) {/tex} is

A

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

B

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

{tex}3{/tex}

D

{tex} \sqrt { 3 } {/tex}

##### Explanation

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Q 4. {tex} \int \frac { x ^ { 2 } - 1 } { x ^ { 3 } \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } d x = {/tex}

A

{tex} \frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { x ^ { 2 } } + c {/tex}

B

{tex} \frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { x ^ { 3 } } + c {/tex}

C

{tex} \frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { x } + c {/tex}

{tex} \frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { 2 x ^ { 2 } } + c {/tex}

##### Explanation

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Q 5. Let {tex} I = \int \frac { e ^ { x } } { e ^ { 4 x } + e ^ { 2 x } + 1 } d x , J = \int \frac { e ^ { - x } } { e ^ { - 4 x } + e ^ { - 2 x } + 1 } d x . {/tex} Then, for an arbitrary constant {tex} C , {/tex} the value of {tex} \mathrm { J } -\mathrm I{/tex} equals

A

{tex} \frac { 1 } { 2 } \log \left( \frac { e ^ { 4 x } - e ^ { 2 x } + 1 } { e ^ { 4 x } + e ^ { 2 x } + 1 } \right) + C {/tex}

B

{tex} \frac { 1 } { 2 } \log \left( \frac { e ^ { 2 x } + e ^ { x } + 1 } { e ^ { 2 x } - e ^ { x } + 1 } \right) + C {/tex}

{tex} \frac { 1 } { 2 } \log \left( \frac { e ^ { 2 x } - e ^ { x } + 1 } { e ^ { 2 x } + e ^ { x } + 1 } \right) + C {/tex}

D

{tex} \frac { 1 } { 2 } \log \left( \frac { e ^ { 4 x } + e ^ { 2 x } + 1 } { e ^ { 4 x } - e ^ { 2 x } + 1 } \right) + C {/tex}

##### Explanation

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Q 6. The area between the curve , the -axis and the ordinates of two minima of the curve is

sq unit

B

sq unit

C

sq unit

D

sq unit

##### Explanation

For maxima or minima, put , we get

Then,
and
Required area

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Q 7. The area bounded by the curves and is

A

2 sq unit

B

sq unit

sq unit

D

sq unit

##### Explanation

Area of square sq unit

Area of circle sq unit
Required area sq unit

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Q 8. The area of the closed igure bounded by and and the -axis is

sq unit

B

sq unit

C

sq unit

D

sq unit

##### Explanation

Required area

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Q 9. The area of the region by curves is

A

B

D

None of these

##### Explanation

Required area

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Q 10. The area bounded by the parabola and the line and is

A

B

C

##### Explanation

Required area area of curve PSRQP

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Q 11. The areas of the figure into which curve divides the circle are in the ratio

A

B

D

None of these

##### Explanation

We have,
Area bounded by the two curves

Area bounded by and outside

Required ratio

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Q 12. Area bounded by the curve and -axis is

1 sq unit

B

1/2 sq unit

C

2 sq unit

D

None of these

##### Explanation

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Q 13. The slope of tangent to a curve at is . If the curve passes through the point (1, 2), then the area of the region bounded by the curve, the -axis and the line is

sq unit

B

sq unit

C

sq unit

D

6 sq unit

##### Explanation

We have,
, it passes through (1, 2)

Then,
Required area

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Q 14. The area common to the circle and the parabola is

A

sq unit

B

sq unit

sq unit

D

None of these

##### Explanation

Given equation of curves are
and
The point of intersection are

Required area,
area of curve
[area of curve area of curve ]

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Q 15. is equal to

A

B

D

##### Explanation

Let

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Q 16. Let . Then, is

Continuous and non differentiable at

B

Discontinuous at

C

Neither continuous nor differentiable at

D

Non-differentiable at

##### Explanation

We have,

Clearly, is everywhere continuous and differentiable except at

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Q 17. is equal to

A

0

B

C

We have,