Indefinite Integration — Chapter Test | Class 12 IMO

Q1

If f(x) is an improper rational function like (x² + 1)/(x² - 5x + 6), what is the first step in integration?

For improper rational functions (degree of numerator ≥ denominator), we must perform long division to express it as a polynomial plus a proper rational function.

Q2

A manufacturing plant in Noida finds its marginal cost of producing x units is given by MC = 3x² - 2x + 5. If the fixed cost (cost at x=0) is ₹ 5000, find the total cost function C(x).

C(x) = ∫ (3x² - 2x + 5) dx = x³ - x² + 5x + K. Since C(0) = 5000, K = 5000.

Q3

A Vande Bharat express train accelerating out of New Delhi station has an acceleration a(t) = 2 m/s². If its initial velocity is 10 m/s, what is its velocity function v(t)?

v(t) = ∫ a(t) dt = ∫ 2 dt = 2t + C. v(0) = 10 implies C = 10.

Q4

What is the process of finding a function whose derivative is given?

Integration is the inverse process of differentiation. If d/dx f(x) = g(x), then the integral of g(x) is f(x).

Q5

In the expression ∫ f(x) dx, what is the term 'f(x)' called?

The function f(x) which is to be integrated is called the integrand.

Q6

∫sin²x dx equals:

Using identity sin²x = (1 − cos 2x)/2. ∫(1 − cos 2x)/2 dx = x/2 − (sin 2x)/4 + C.

Q7

Evaluate: ∫ dx / ((x-1)(x-2))

Using partial fractions 1/(x-2) - 1/(x-1), integration yields log|x-2| - log|x-1| = log|(x-2)/(x-1)| + C.

Q8

Using partial fractions, ∫(dx)/(x² − 1) equals:

1/(x²−1) = 1/[(x−1)(x+1)] = A/(x−1) + B/(x+1). Solving: A=1/2, B=−1/2. Integral = (1/2)ln|x−1| − (1/2)ln|x+1| + C = (1/2)ln|(x−1)/(x+1)| + C.

Q9

A startup's customer base grows at a rate of dN/dt = 100e^(0.1t) users per month. If they start with 1000 users at t=0, how many users will they have at time t?

N(t) = ∫ 100e^(0.1t) dt = 1000e^(0.1t) + C. N(0) = 1000 => 1000(1) + C = 1000 => C = 0.

Q10

Evaluate: ∫ dx / √(x² - 4)

Standard form ∫ dx / √(x² - a²) = log|x + √(x² - a²)| + C. Here a = 2.

Q11

Evaluate ∫(x² + 1)/(x² − 1) dx.

(x²+1)/(x²−1) = 1 + 2/(x²−1). ∫1 dx + 2∫dx/(x²−1) = x + 2[(1/2)ln|(x−1)/(x+1)|] + C = x + ln|(x−1)/(x+1)| + C.

Q12

Evaluate: ∫ 1 / (sin⁴ x + cos⁴ x) dx

Divide by cos⁴ x to get sec² x dx / (tan⁴ x + 1). Put tan x = t. Then ∫ (t² + 1 - t²) / (t⁴ + 1) dt. Breaking and substituting u = t - 1/t gives the result.

Q13

The ILATE rule for integration by parts prioritizes functions in which order?

ILATE stands for Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential — this is the priority order for choosing the first function u.

Q14

∫cos²x dx is equal to:

cos²x = (1 + cos 2x)/2. Integral = ∫(1/2)dx + (1/2)∫cos 2x dx = x/2 + (sin 2x)/4 + C.

Q15

Evaluate: ∫ (eˣ (1 + x log x) / x) dx

Rewriting as ∫ eˣ (1/x + log x) dx. This matches ∫ eˣ (f'(x) + f(x)) dx where f(x) = log x. Result is eˣ f(x) = eˣ log x + C.