NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Exercise 7.3

Flashcard for Question 1

Short Formula Recall:
sin²θ = (1 − cos 2θ) / 2

Quick Tip:
Use the half-angle identity to simplify before integrating.

Common Mistake:
Trying to integrate sin² directly without using the identity.

Exam Insight:
A standard integral simplification problem, commonly appearing in the integration-by-identities section.

Flashcard for Question 2

Short Formula Recall:
sin A cos B = ½[sin(A + B) + sin(A − B)]

Quick Tip:
Apply the product-to-sum identity before integrating.

Common Mistake:
Directly trying to integrate the product without converting to a sum.

Exam Insight:
Classic example from trigonometric integration identities — often worth 2 marks in board exams.

Flashcard for Question 3

Short Formula Recall:
cos A cos B = ½[cos(A + B) + cos(A − B)]

Quick Tip:
Reduce the product step-by-step using product-to-sum formulas until only single cosine terms remain for easy integration.

Common Mistake:
Skipping intermediate steps in product-to-sum conversion and making sign errors in angles.

Exam Insight:
A slightly tricky integration identity problem – good practice for multi-angle product simplifications.

Flashcard for Question 4

Short Formula Recall:
sin³θ = sinθ (1 − cos²θ)

Quick Tip:
Split sin³ into sinθ(1 − cos²θ) and use u-substitution with u = cos(2x + 1).

Common Mistake:
Forgetting the chain rule factor when substituting, leading to a missing coefficient in the final answer.

Exam Insight:
Common example for testing odd-power sine integration and substitution skills.

Flashcard for Question 8

Short Formula Recall:
1 − cos x = 2 sin²(x/2), 1 + cos x = 2 cos²(x/2) ⇒ (1 − cos x)/(1 + cos x) = tan²(x/2)

Quick Tip:
Convert to half-angle form and use tan²θ = sec²θ − 1 to integrate easily.

Common Mistake:
Forgetting to apply the half-angle substitution correctly or missing the factor from dx when substituting u = x/2.

Exam Insight:
A standard trigonometric identity simplification before integration — shows up often in board exam practice papers.

Flashcard for Question 9

Short Formula Recall:
1 + cos x = 2 cos²(x/2), cos x = 1 − 2 sin²(x/2)

Quick Tip:
Divide numerator and denominator by cos²(x/2) to simplify, then integrate using standard sec² and tan formulas.

Common Mistake:
Trying direct substitution without simplifying – leads to messy algebra.

Flashcard for Question 10

Short Formula Recall:
sin²x = (1 − cos 2x)/2 ⇒ sin⁴x = [(1 − cos 2x)/2]²

Quick Tip:
Use the power-reduction identity twice to convert to a sum of cosines for easy integration.

Common Mistake:
Forgetting to square the (1/2) factor when expanding sin²x.

Flashcard for Question 11

Quick Tip:
Use the power-reduction identity two times, then integrate each term independently, being careful to include the inner derivative of 2x.

Common Mistake:
Forgetting the chain rule factor (1/2) when integrating functions of 2x, 4x, etc.

Exam Insight:
A straightforward application of multiple-angle identities — worth 2–3 marks if done neatly

Flashcard for Question 12

Quick Tip:
Cancel the (1 + cos x) term, leaving 1 − cos x, which integrates easily.

Common Mistake:
Forgetting to simplify before integrating — trying direct integration makes it unnecessarily hard.

Exam Insight:
A classic simplification-before-integration problem — often appears in short-answer sections.

Flashcard for Question 13

Short Formula Recall:
cosA − cosB = −2 sin((A + B)/2) sin((A − B)/2);
sin2θ = 2 sinθ cosθ;

Quick Tip:
Use sum-to-product to simplify the fraction first — it reduces to a simple sum of cosines.

Common Mistake:
Skipping the cancellation step after converting to product form and making algebraic sign errors.

Exam Insight:
A neat 2–3 mark question: simplify first (reduces to elementary integrals) rather than attempting direct integration.

Flashcard for Question 17

Quick Tip:
Split the fraction: (sin³+cos³)/(sin² cos²) = sin/cos² + cos/sin² and match each term to a known derivative.

Common Mistake:
Not splitting the integrand or mishandling the sign for the csc derivative (cos/sin² = −(d/dx)csc x).

Exam Insight:
Recognising derivatives of sec and csc gives a one-step integral – typical 1–2 mark question if simplified neatly.

Flashcard for Question 18

Quick Tip:
Substitute cos 2x and notice cancellation with 1-2 sin²x — the numerator becomes 1.

Common Mistake:
Expanding tan²x too early instead of spotting that the numerator simplifies directly.

Exam Insight:
A quick simplification turns the problem into ∫ sec²x dx, which is a one-step standard integral.

Flashcard for Question 21

Short Formula Recall:
sin⁻¹(cos x) = π/2 − x, for x ∈ [0, π]

Quick Tip:
Replace sin⁻¹(cos x) with its equivalent linear form to turn the integral into a basic polynomial integration.

Common Mistake:
Forgetting the principal value range of sin⁻¹, leading to wrong simplification outside [0, π].

Flashcard for Question 22

Exam Insight:
This is a standard trick-question – simplify first using sum-to-product then use the tangent half-angle; it’s worth 3-4 marks if shown neatly.