#3 Evaluate the Integral - Edubirdie

University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

393

Pages:

3
Author:

Shahbaz A.

Integration shahbaz ahmed August 2024 ’ Relevant concepts and formulas limn−→∞ (1 + n1 )n = e Putting x= 1 n When n −→ ∞ =⇒ x → 0 Also x= 1 n =⇒ n = 1 x When x → 0 =⇒ n → ∞ So 1 limx→0 (1 + x) x = e Where e is an irrational number such that 2 < e < 3 Also 1 loge x = ln x d dx (ln x) R 1 x = 1 x dx = ln |x| + C ......................................... Evaluate the integral Z cos x dx sin x ln sin x Z cos x dx sin x ln sin x Solution Sine d dx (ln sin x) = 1 d sin x dx (sin x) d dx (ln sin x) = 1 sin x d(ln sin x) = cos x sin x cos x dx Putting in equation (1) Z cos x dx = sin x ln sin x 2 Z d(ln sin x) ln sin x (1) Z cos x dx = ln[ln | sin x|] + C sin x ln sin x 3

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