Advertisement Remove all ads

Evaluate the Following Integrals: ∫ E 2 X ( 1 − Sin 2 X 1 − Cos 2 X ) D X - Mathematics

Advertisement Remove all ads
Advertisement Remove all ads
Advertisement Remove all ads
Sum

Evaluate the following integrals:

\[\int e^{2x} \left( \frac{1 - \sin2x}{1 - \cos2x} \right)dx\]
Advertisement Remove all ads

Solution

\[\text{ We have,} \]

\[I = \int e^{2x} \left( \frac{1 - \sin2x}{1 - \cos2x} \right)dx\]

\[ = \int e^{2x} \left( \frac{1 - 2 sinx \cos x}{2 \sin^2 x} \right)dx\]

\[\text{ Put  t }= 2x . \text{ Then dt} = 2dx\]

\[\text{ Therefore }, \]

\[I = \frac{1}{2}\int e^t \left( \frac{1 - 2 \sin\frac{t}{2} \cos\frac{t}{2}}{2 \sin^2 \frac{t}{2}} \right)dt\]

\[ = \frac{1}{4}\int e^t \left( \frac{1 - 2 \sin\frac{t}{2} \cos\frac{t}{2}}{\sin^2 \frac{t}{2}} \right)dt\]

\[ = \frac{1}{4}\int e^t \left( \frac{1}{\sin^2 \frac{t}{2}} - \frac{2 \sin\frac{t}{2}\cos\frac{t}{2}}{\sin^2 \frac{t}{2}} \right)dt\]

\[ = \frac{1}{4}\int e^t \left( {cosec}^2 \frac{t}{2} - 2\cot\frac{t}{2} \right)dt\]

\[ = - \frac{1}{4}\int e^t \left( 2\cot\frac{t}{2} - {cosec}^2 \frac{t}{2} \right)dt\]

\[\text{ Consider, }f\left( x \right) = 2\cot\frac{t}{2}, \text{ then f}^ \left( x \right) = - {cosec}^2 \frac{t}{2}\]

\[ \text{Thus, the given integrand is of the form} \text{ e}^x \left[ f   \left( x \right) + f^{ '} \left( x \right) \right] . \]

\[\text{ Therefore, I }= - \frac{1}{4}\left( 2\cot\frac{t}{2} \right) e^t + c\]

\[ = - \frac{1}{4}\left( 2\cot\frac{2x}{2} \right) e^{2x} + c\]

\[\text{ Hence, }\int e^{2x} \left( \frac{1 - \sin2x}{1 - \cos2x} \right)dx = - \frac{1}{2}\left( \cot x \right) e^{2x} + c\]

Concept: Evaluation of Simple Integrals of the Following Types and Problems
  Is there an error in this question or solution?

APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Exercise 19.26 | Q 24 | Page 143
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×