Advertisements
Advertisements
प्रश्न
उत्तर
\[\text{ Let I } = \int\frac{dx}{\sqrt{x^2 - a^2}}\]
\[\text{ Putting x} = a \tan \theta\]
\[ \Rightarrow dx = a \sec^2 \text{ θ dθ }\]
\[ \therefore I = \int\frac{a \cdot se c^2\text{ θ dθ }}{\sqrt{a^2 \tan^2 \theta + a^2}}\]
\[ = \int\frac{a \sec^2 \theta \cdot d\theta}{a\sqrt{1 + \tan^2 \theta}}\]
\[ = \int\frac{\sec^2 \theta \cdot \text{ dθ }}{\sec\theta}\]
\[ = \int\sec\theta \cdot d\theta\]
\[ = \int\sec\theta \cdot d\theta\]
\[ = \text{ ln } \left| \sec\theta + \tan\theta \right| + C\]
\[ = \text{ ln }\left| \sec\theta + \sqrt{\sec^2 \theta - 1} \right| + C\]
\[ = \text{ ln }\left| \frac{x}{a} + \sqrt{\frac{x^2}{a^2} - 1} \right| + C\]
\[ = \text{ ln} \left| x + \sqrt{x^2 - a^2} \right| - \ln a + C\]
\[ = \text{ ln }\left| x + \sqrt{x^2 - a^2} \right| + C'\]
\[\text{ where C' = C - ln a }\]
APPEARS IN
संबंधित प्रश्न
\[\int\left\{ x^2 + e^{\log x}+ \left( \frac{e}{2} \right)^x \right\} dx\]
If f' (x) = x + b, f(1) = 5, f(2) = 13, find f(x)
\[\int\frac{1}{4 \sin^2 x + 4 \sin x \cos x + 5 \cos^2 x} \text{ dx }\]
\[ \int\left( 1 + x^2 \right) \ \cos 2x \ dx\]
\[\int \left( e^x + 1 \right)^2 e^x dx\]
