Advertisements
Advertisements
Question
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
Sum
Advertisements
Solution
\[\int\left( \frac{2x + 3}{\left( x - 1 \right)^2} \right)dx\]
\[ = \int\left[ \frac{2x - 2 + 2 + 3}{\left( x - 1 \right)^2} \right]dx\]
\[ = \int\left[ \frac{2\left( x - 1 \right) + 5}{\left( x - 1 \right)^2} \right]dx\]
\[ = 2\int\frac{dx}{\left( x - 1 \right)} + 5\int \left( x - 1 \right)^{- 2} dx\]
\[ = \text{2 ln }\left| x - 1 \right| + 5\left[ \frac{\left( x - 1 \right)^{- 2 + 1}}{- 2 + 1} \right] + C\]
\[ = \text{2 ln }\left| x - 1 \right| - \frac{5}{x - 1} + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\left( 2^x + \frac{5}{x} - \frac{1}{x^{1/3}} \right)dx\]
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]
\[\int \left( 3x + 4 \right)^2 dx\]
\[\int \cos^{- 1} \left( \sin x \right) dx\]
Write the primitive or anti-derivative of
\[f\left( x \right) = \sqrt{x} + \frac{1}{\sqrt{x}} .\]
\[\int\frac{1}{\sqrt{2x + 3} + \sqrt{2x - 3}} dx\]
\[\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx\]
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
\[\int\frac{x^2 + x + 5}{3x + 2} dx\]
\[\int \sin^2 \frac{x}{2} dx\]
\[\int\frac{e^x + 1}{e^x + x} dx\]
\[\int\frac{- \sin x + 2 \cos x}{2 \sin x + \cos x} dx\]
\[\int\frac{2 \cos 2x + \sec^2 x}{\sin 2x + \tan x - 5} dx\]
\[\int \sin^5\text{ x }\text{cos x dx}\]
\[\int\frac{1}{\left( x + 1 \right)\left( x^2 + 2x + 2 \right)} dx\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} dx\]
\[\int\frac{x^4 + 1}{x^2 + 1} dx\]
\[\int\frac{2x + 5}{x^2 - x - 2} \text{ dx }\]
\[\int\frac{1}{\left( \sin x - 2 \cos x \right)\left( 2 \sin x + \cos x \right)} \text{ dx }\]
\[\int\frac{2 \sin x + 3 \cos x}{3 \sin x + 4 \cos x} dx\]
\[\int x^2 \sin^{- 1} x\ dx\]
\[\int x \sin^3 x\ dx\]
\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]
\[\int\frac{1}{x \log x \left( 2 + \log x \right)} dx\]
\[\int\frac{x^2 + x - 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx\]
\[\int\frac{x^2 + 1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} dx\]
\[\int\frac{1}{\left( x^2 + 1 \right) \sqrt{x}} \text{ dx }\]
The value of \[\int\frac{\sin x + \cos x}{\sqrt{1 - \sin 2x}} dx\] is equal to
\[\int\frac{1}{e^x + 1} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{\sin x}{\sqrt{\cos^2 x - 2 \cos x - 3}} \text{ dx }\]
\[\int\frac{1}{\left( \sin x - 2 \cos x \right) \left( 2 \sin x + \cos x \right)} \text{ dx }\]
\[\int\frac{\sin^6 x}{\cos x} \text{ dx }\]
\[\int\frac{1}{\sec x + cosec x}\text{ dx }\]
\[\int\left( 2x + 3 \right) \sqrt{4 x^2 + 5x + 6} \text{ dx}\]
\[\int\frac{x}{x^3 - 1} \text{ dx}\]
\[\int\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} \text{ dx}\]
