Advertisements
Advertisements
प्रश्न
उत्तर
\[\int \left( \log x \right)_{}^2 {x \cdot} dx\]
` "Taking log x"^2" as the first function and x as the second function ". `
\[ = \left( \log x \right)^2 \int xdx - \int\left\{ \frac{d}{dx} \left( \log x \right)^2 \int x\ dx \right\}dx\]
\[ = \left( \log x \right)^2 \cdot \frac{x^2}{2} - \int\frac{\left( 2 \log x \right)}{x} \times \frac{x^2}{2} dx\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \int x_{II} \log x_I dx\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \left[ \log x \int x\ dx - \int\left\{ \frac{d}{dx}\left( \log x \right)\int x\ dx \right\}dx \right]\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \left[ \log x \cdot \frac{x^2}{2} - \int\frac{1}{x} \times \frac{x^2}{2}dx \right]\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \log x \cdot \frac{x^2}{2} + \frac{x^2}{4} + C\]
\[ = \frac{x^2}{2}\left[ \left( \log x \right)^2 - \log x + \frac{1}{2} \right] + C\]
APPEARS IN
संबंधित प्रश्न
` ∫ sin x \sqrt (1-cos 2x) dx `
Evaluate the following integrals:
The primitive of the function \[f\left( x \right) = \left( 1 - \frac{1}{x^2} \right) a^{x + \frac{1}{x}} , a > 0\text{ is}\]
\[\int\text{ cos x cos 2x cos 3x dx}\]
\[\int \sec^4 x\ dx\]
\[\int\frac{1}{2 + \cos x} \text{ dx }\]
Find : \[\int\frac{dx}{\sqrt{3 - 2x - x^2}}\] .
