Integrate the following functions w.r.t. x : ex.log(sinex)tan(ex) - Mathematics and Statistics

Question

Integrate the following functions w.r.t. x : $e^{x} . \frac{\log ({\sin e}^{x})}{\tan (e^{x})}$

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Solution

Let I = $\int \frac{e^{x} . \log ({\sin e}^{x})}{\tan (e^{x})} . d x$

= $\int \log ({\sin e}^{x}) . e^{x} . \cot (e^{x}) d x$

Put log (sin e^x) = t

∴ $\frac{1}{\sin (e^{x})} . \cos (e^{x}) . e^{x} d x$ = dt

∴ e^x . cot (e^x) dx = dt

∴ I = $\int t d t = \frac{t^{2}}{2} + c$

= $\frac{1}{2} {[\log ({\sin e}^{x})]}^{2} + c$ .

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Methods of Integration: Integration by Substitution

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Chapter 3: Indefinite Integration - Exercise 3.2 (A) [Page 110]

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Balbharati Mathematics and Statistics 2 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board

Chapter 3 Indefinite Integration
Exercise 3.2 (A) | Q 1.07 | Page 110

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Integrate the following functions w.r.t. x : ex.log(sinex)tan(ex) - Mathematics and Statistics

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