# Evaluate-I-ab-sin-axcos-bxdx-if-a-b-and-use-it-to-0-n-sin-3xcos-2xdx-3-3-5-

Question Number 76965 by peter frank last updated on 02/Jan/20
$${Evaluate} \\$$$${I}_{{ab}} =\int\mathrm{sin}\:{ax}\mathrm{cos}\:{bxdx} \\$$$${if}\:{a}\neq{b}\:{and}\:{use}\:{it}\:{to} \\$$$$\int_{\mathrm{0}} ^{{n}} \mathrm{sin}\:\mathrm{3}{x}\mathrm{cos}\:\mathrm{2}{xdx}=\frac{\mathrm{3}−\sqrt{\mathrm{3}}}{\mathrm{5}} \\$$
Answered by mr W last updated on 01/Jan/20
$$\mathrm{sin}\:\left({ax}+{bx}\right)=\mathrm{sin}\:{ax}\:\mathrm{cos}\:{bx}+\mathrm{cos}\:{ax}\:\mathrm{sin}\:{bx} \\$$$$\mathrm{sin}\:\left({ax}−{bx}\right)=\mathrm{sin}\:{ax}\:\mathrm{cos}\:{bx}−\mathrm{cos}\:{ax}\:\mathrm{sin}\:{bx} \\$$$$\Rightarrow\mathrm{sin}\:{ax}\:\mathrm{cos}\:{bx}=\frac{\mathrm{sin}\:\left({a}+{b}\right){x}+\mathrm{sin}\:\left({a}−{b}\right){x}}{\mathrm{2}} \\$$$${I}=\int\mathrm{sin}\:{ax}\:\mathrm{cos}\:{bx}\:{dx} \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\left(\mathrm{sin}\:\left({a}+{b}\right){x}+\mathrm{sin}\:\left({a}−{b}\right){x}\right){dx} \\$$$$=−\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\mathrm{cos}\:\left({a}+{b}\right){x}}{{a}+{b}}+\frac{\mathrm{cos}\:\left({a}−{b}\right){x}}{{a}−{b}}\right]+{C} \\$$
Commented by peter frank last updated on 02/Jan/20
$${thank}\:{you} \\$$
Commented by peter frank last updated on 02/Jan/20
$${how}\:{about}\:{part}\:{b} \\$$
Commented by mr W last updated on 02/Jan/20
$${i}\:{think}\:{you}\:{should}\:{be}\:{able}\:{to}\:{do} \\$$$${part}\:{b}\:{by}\:{yourself}.\:{but}\:{i}\:{can}\:{not}, \\$$$${because}\:{i}\:{don}'{t}\:{know}\:{what}\:{is}\:{n}\:{in} \\$$$${your}\:{question}\:\int_{\mathrm{0}} ^{\:{n}} . \\$$
Commented by peter frank last updated on 02/Jan/20
$${thank}\:{you} \\$$