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Write a Value of ∫ Sin X Cos 3 X D X - Mathematics

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Sum

Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]

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Solution

\[\text{ Let I }= \int \frac{\sin x}{\cos^3 x}dx\]
\[\text{ Let  cos x  }= t\]
\[ \Rightarrow - \text{ sin x dx} = dt\]
\[ \Rightarrow \text{ sin x dx }= - dt\]
\[ \therefore I = - \int \frac{dt}{t^3}\]
\[ = - \int t^{- 3} dt\]
\[ = - \left[ \frac{t^{- 3 + 1}}{- 3 + 1} \right] + C\]
\[ = \frac{1}{2 t^2} + C\]
\[ = \frac{1}{2 \cos^2 x} + C \left( \because t = \cos x \right)\]
\[ = \frac{1}{2} \text{ sec}^2 x + C\]

Concept: Methods of Integration: Integration by Substitution
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Very Short Answers | Q 28 | Page 197
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