Answer the questions below

MATHEMATICS · Subjective · A Level · Mathematics · Integration · Integration

Problem

Answer the questions below. (i) Derive general formulae for ∫sin⁡(ax+b) dx\displaystyle \int \sin(ax+b)\,dx and ∫cos⁡(ax+b) dx\displaystyle \int \cos(ax+b)\,dx where a≠0a\ne 0. (ii) Using your result, evaluate I(a)=∫0πsin⁡(ax) dxI(a)=\displaystyle \int_{0}^{\pi}\sin(ax)\,dx and simplify the answer. (iii) Hence, find lim⁡a→0I(a)\displaystyle \lim_{a\to 0} I(a).

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