Advertisements
Advertisements
Question
\[\int\left( x + 1 \right) \text{ log x dx }\]
Sum
Advertisements
Solution
\[\int \left( x + 1 \right)_{II} . \log_1 \text{ x dx }\]
\[ = \log x\int\left( x + 1 \right)dx - \int\left\{ \frac{d}{dx}\left( \log x \right)\int\left( x + 1 \right)dx \right\}dx\]
\[ = \log x\left[ \frac{x^2}{2} + x \right] - \int \frac{1}{x}\left( \frac{x^2}{2} + x \right)dx\]
\[ = \log x\left( \frac{x^2}{2} + x \right) - \int \left( \frac{x}{2} + 1 \right)dx\]
\[ = \log x\left( \frac{x^2}{2} + x \right) - \left( \frac{x^2}{4} + x \right) + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int \left( 3x + 4 \right)^2 dx\]
\[\int\left( \sec^2 x + {cosec}^2 x \right) dx\]
\[\int \cos^2 \text{nx dx}\]
\[\int\frac{\sin 2x}{\left( a + b \cos 2x \right)^2} dx\]
\[\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} dx\]
` ∫ sec^6 x tan x dx `
\[\int \sin^5 x \cos x \text{ dx }\]
\[\int\frac{1}{\sin x \cos^3 x} dx\]
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 - 1}} dx\]
` ∫ { x^2 dx}/{x^6 - a^6} dx `
\[\int\frac{x}{\sqrt{4 - x^4}} dx\]
\[\int\frac{2x}{2 + x - x^2} \text{ dx }\]
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} dx\]
\[\int\frac{x + 1}{\sqrt{4 + 5x - x^2}} \text{ dx }\]
\[\int\frac{x + 1}{\sqrt{x^2 + 1}} dx\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{\cos x}{\cos 3x} \text{ dx }\]
\[\int\frac{1}{1 + 3 \sin^2 x} \text{ dx }\]
\[\int e^\sqrt{x} \text{ dx }\]
\[\int \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ dx }\]
\[\int x^2 \sin^{- 1} x\ dx\]
\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{x}{\left( x + 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{x^3 - 1}{x^3 + x} dx\]
\[\int\frac{1}{\sin x \left( 3 + 2 \cos x \right)} dx\]
Write a value of
\[\int e^{3 \text{ log x}} x^4\text{ dx}\]
\[\int\frac{\cos2x - \cos2\theta}{\cos x - \cos\theta}dx\] is equal to
\[\int\frac{x^2}{\left( x - 1 \right)^3} dx\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{1}{\sqrt{3 - 2x - x^2}} \text{ dx}\]
\[\int\frac{1}{a + b \tan x} \text{ dx }\]
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
\[\int\frac{\sin^6 x}{\cos x} \text{ dx }\]
\[\int \tan^5 x\ \sec^3 x\ dx\]
\[\int\frac{\log \left( \log x \right)}{x} \text{ dx}\]
\[\int\frac{\log \left( 1 - x \right)}{x^2} \text{ dx}\]
\[\int\frac{1}{1 + x + x^2 + x^3} \text{ dx }\]
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
