Pure: Indices and Logarithms

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Grade 1 · Term 1

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Revise Pure: Indices and Logarithms fast with concise, exam-ready key points

About A/L Combined Mathematics: Combined Mathematics consists of Pure Mathematics and Applied Mathematics across two years.

• Rewrite as ∫(sin x / cos x) dx. 3. Let u = cos x, then du = -sin x dx, so sin x dx = -du. 4. Substitute: ∫(-1/u) du = -ln|u| + C. 5. Substitute back: -ln|cos x| + C. ▶ Integral of cot x
• Rewrite as ∫(cos x / sin x) dx. 3. Let u = sin x, then du = cos x dx. 4. Substitute: ∫(1/u) du = ln|u| + C. 5. Substitute back: ln|sin x| + C. ▶ Integral of sec x
• Multiply numerator and denominator by (cosec x + cot x): ∫[cosec x (cosec x + cot x)] / (cosec x + cot x) dx = ∫(cosec²x + cosec x cot x) / (cosec x + cot x) dx. 3. Let u = cosec x + cot x. Then du = (-cosec x cot x - cosec²x) dx = -(cosec²x + cosec x cot x) dx. 4. Substitute: ∫(-1/u) du = -ln|u| + C. 5. Substitute back: -ln|cosec x + cot x| + C. ▶ Integral of sin²x
• Use identity sin²x = (1 - cos2x)/2. 3. ∫(1/2 - (1/2)cos2x) dx = (1/2)∫1 dx - (1/2)∫cos2x dx. 4. = (1/2)x - (1/2)(sin2x / 2) + C = (1/2)x - (1/4)sin2x + C. ▶ Integral of cos²x
• Use identity cos²x = (1 + cos2x)/2. 3. ∫(1/2 + (1/2)cos2x) dx = (1/2)∫1 dx + (1/2)∫cos2x dx. 4. = (1/2)x + (1/2)(sin2x / 2) + C = (1/2)x + (1/4)sin2x + C. ▶ Integral of tan²x

Source: Idasara knowledge pack — english/AL_SCIENCE/combined_mathematics v13, short_notes: Lesson 10 - Competency 6 _ Manipulates laws of indices and laws of logarithms.

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