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Question Number 10241 by FilupSmith last updated on 31/Jan/17

$${\int }_a^{a+\delta }\frac{\sin \left(x\right)}{\cos (x+\delta )}dx=???$$

Commented by prakash jain last updated on 31/Jan/17

$$\frac{\sin \left(x\right)}{\cos (x+\delta )}=\frac{\sin (x+\delta -\delta )}{\cos (x+\delta )}$$$$=\frac{\sin (x+\delta )\cos \delta -\cos (x+\delta )\sin \delta }{\cos (x+\delta )}$$$$=\tan (x+\delta )\cos \delta -\sin \delta$$$${\int }_a^{a+\delta }(\tan (x+\delta )\cos \delta -\sin \delta )dx$$$$={[-\cos \delta \ln \cos (x+\delta )-x\sin \delta ]}_a^{a+\delta }$$$$=[-\cos \delta \ln \frac{\cos (a+\delta )}{\cos a}-\delta \sin \delta ]$$

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