Question 1

Passage: Let \,d (2+ )(3+4 ) =A |2+ |+B |3+4 |. Question: What is the value of A ?

Accepted Answer

Let I = \,d (2+ )(3+4 ) Substitute = t, which gives - \,d = dt \,d = -dt I = -dt (2+t)(3+4t) Using partial fractions: -1 (t+2)(4t+3) = P t+2 + Q 4t+3 -1 = P(4t+3) + Q(t+2) Substituting t = -2, we get -1 = P(-8+3) P = 1 5 Substituting t = - 3 4 , we get -1 = Q (- 3 4 +2 ) Q = - 4 5 Therefore, the integral becomes: I = ( 1/5 t+2 - 4/5 4t+3 ) dt I = 1 5 |t+2| - 4 5 1 4 |4t+3| + C I = 1 5 |t+2| - 1 5 |4t+3| + C Substituting back t = : I = 1 5 |2+ | - 1 5 |3+4 | + C Comparing this with A |2+ | + B |3

Question 2

Passage: Let \,d (2+ )(3+4 ) =A |2+ |+B |3+4 |. Question: What is the value of B ?

Accepted Answer

Let I = \,d (2+ )(3+4 ) . Substitute t = , which gives dt = - \,d . The integral becomes: I = -dt (2+t)(3+4t) Using partial fractions, we can write: -1 (2+t)(3+4t) = A' 2+t + B' 3+4t -1 = A'(3+4t) + B'(2+t) Putting t = -2, we get -1 = A'(3 - 8) A' = 1 5 . Putting t = - 3 4 , we get -1 = B' (2 - 3 4 ) -1 = B' ( 5 4 ) B' = - 4 5 . So, the integral is: I = ( 1/5 2+t - 4/5 3+4t ) dt I = 1 5 |2+t| - 4 5 1 4 |3+4t| + C I = 1 5 |2+t| - 1 5 |3+4t| + C Substituting back t = : I = 1 5 |2+ | - 1 5 |3+4 | +

Question 3

What is dx 10^x + 10^ -x equal to?

Accepted Answer

Let I = dx 10^x + 10^ -x I = 10^x 10^ 2x + 1 dx Let 10^x = t 10^x 10 \, dx = dt 10^x \, dx = dt 10 I = 1 10 dt t^2 + 1 I = 1 10 ^ -1 (t) + c I = 1 10 ^ -1 (10^x) + c Answer: 1 10 ^ -1 (10^x) + c

Question 4

Consider the following for the two (02) items that follow: Let the function y = (1 - x)^ -1 , where x 2n and n is an integer. What is y \, dx equal to?

Accepted Answer

Given y = (1 - x)^ -1 y = 1 1 - x Using the half-angle formula 1 - x = 2 ^2(x/2) y = 1 2 ^2(x/2) = 1 2 ^2(x/2) Integrating with respect to x: y \, dx = 1 2 ^2(x/2) \, dx y \, dx = 1 2 ( - (x/2) 1/2 ) + c y \, dx = - (x/2) + c Answer: - (x/2) + c

Question 5

For the following two (02) items: Let u = e^x x\, dx and v = e^x x\, dx. Consider the following: I. du dx = -v II. dv dx = -u Which of the above is/are correct?

Accepted Answer

Given u = e^x x\, dx and v = e^x x\, dx. Differentiating both sides with respect to x, we get: du dx = e^x x dv dx = e^x x Evaluating the integrals, we have: u = e^x 2 ( x + x) + C_1 v = e^x 2 ( x - x) + C_2 Comparing the derivatives with u and v, it is clear that: du dx -v dv dx -u Therefore, neither statement I nor statement II is correct.

Question 6

For the following two (02) items: Let u = e^x x\, dx and v = e^x x\, dx. What is u + v equal to?

Accepted Answer

Given u = e^x x\, dx and v = e^x x\, dx Adding u and v: u + v = e^x ( x + x)\, dx Using the standard integral e^x (f(x) + f'(x))\, dx = e^x f(x), where f(x) = x and f'(x) = x: u + v = e^x x From the given expression for v, differentiating with respect to x: dv dx = d dx ( e^x x\, dx ) = e^x x Therefore, u + v = dv dx Answer: dv dx

Indefinite Integration — Previous Year Questions & Solutions