Question Number 200606 by Calculusboy last updated on 20/Nov/23
Answered by Frix last updated on 20/Nov/23
$$\int\frac{\mathrm{cos}\:{x}\:\mathrm{sin}^{\mathrm{2}} \:{x}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}{dx}\:\overset{{t}={x}−\frac{\pi}{\mathrm{4}}} {=} \\ $$$$=\int\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{2}}−\frac{\mathrm{sin}^{\mathrm{2}} \:{t}}{\mathrm{2}}−\frac{\mathrm{tan}\:{t}}{\mathrm{4}}\right){dt}= \\ $$$$=\frac{{t}}{\mathrm{4}}−\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{4}}−\frac{{t}+\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{4}}+\frac{\mathrm{ln}\:\mathrm{cos}\:{t}}{\mathrm{4}}= \\ $$$$=\frac{\mathrm{ln}\:\mid\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\mid}{\mathrm{4}}−\frac{\left(\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\right)\mathrm{cos}\:{x}}{\mathrm{4}}+{C} \\ $$
Commented by Calculusboy last updated on 21/Nov/23
$$\boldsymbol{{thanks}}\:\boldsymbol{{sir}} \\ $$