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∫ 2 X Sec 3 ( X 2 + 3 ) Tan ( X 2 + 3 ) D X - Mathematics

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Question

\[\int2x \sec^3 \left( x^2 + 3 \right) \tan \left( x^2 + 3 \right) dx\]

Sum

Solution

\[\int2x \sec^3 \left( x^2 + 3 \right) \cdot \tan \left( x^2 + 3 \right) dx\]
\[ = \int \sec^2 \left( x^2 + 3 \right) \cdot \sec \left( x^2 + 3 \right) \cdot \tan \left( x^2 + 3 \right) \cdot \text{2x dx}\]
\[\text{Let }\sec \left( x^2 + 3 \right) = t\]
\[ \Rightarrow \sec \left( x^2 + 3 \right) \cdot \tan \left( x^2 + 3 \right) \cdot 2x = \frac{dt}{dx}\]
\[ \Rightarrow \sec \left( x^2 + 3 \right) \cdot \tan \left( x^2 + 3 \right) \cdot \text{2x dx} = dt\]
\[Now, \int \sec^2 \left( x^2 + 3 \right) \cdot \sec \left( x^2 + 3 \right) \cdot \tan \left( x^2 + 3 \right) \cdot \text{2x dx}\]
\[ = \int t^2 dt\]
\[ = \frac{t^3}{3} + C\]
\[ = \frac{\sec^3 \left( x^2 + 3 \right)}{3} + C\]

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Indefinite Integral Problems

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Chapter 19: Indefinite Integrals - Exercise 19.09 [Page 58]

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RD Sharma Mathematics [English] Class 12

Chapter 19 Indefinite Integrals
Exercise 19.09 | Q 39 | Page 58

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