Advertisements
Advertisements
प्रश्न
\[\int\frac{\sqrt{16 + \left( \log x \right)^2}}{x} \text{ dx}\]
योग
उत्तर
\[\text{ Let I }= \int\sqrt{\frac{16 + \left( \log x \right)^2}{x}}\text{ dx}\]
\[\text{ Putting log x }= t\]
\[ \Rightarrow \frac{1}{x} \text{ dx}= dt\]
\[ \therefore I = \int\sqrt{16 + t^2}dt\]
\[ = \int\sqrt{4^2 + t^2}dt\]
\[ = \frac{t}{2} \sqrt{4^2 + t^2} + \frac{4^2}{2} \text{ log} \left| t + \sqrt{4^2 + t^2} \right| + C\]
\[ = \frac{\log x}{2} \sqrt{16 + \left( \log x \right)^2} + 8 \text{ log }\left| \log x + \sqrt{16 + \left( \log x \right)^2} \right| + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\left( 2 - 3x \right) \left( 3 + 2x \right) \left( 1 - 2x \right) dx\]
\[\int\sqrt{x}\left( x^3 - \frac{2}{x} \right) dx\]
\[\int \left( \tan x + \cot x \right)^2 dx\]
\[\int\frac{\cos x}{1 + \cos x} dx\]
\[\int\frac{a}{b + c e^x} dx\]
` ∫ {"cosec" x }/ { log tan x/2 ` dx
\[\int\frac{x + 1}{x \left( x + \log x \right)} dx\]
\[\int\frac{\left( x + 1 \right) e^x}{\sin^2 \left( \text{x e}^x \right)} dx\]
\[\int\frac{e^{m \tan^{- 1} x}}{1 + x^2} dx\]
\[\int\frac{1}{\sqrt{x} + x} \text{ dx }\]
\[\int \sin^7 x \text{ dx }\]
Evaluate the following integrals:
\[\int\cos\left\{ 2 \cot^{- 1} \sqrt{\frac{1 + x}{1 - x}} \right\}dx\]
\[\int\frac{1}{\sqrt{a^2 + b^2 x^2}} dx\]
\[\int\frac{\cos x}{\sqrt{4 + \sin^2 x}} dx\]
\[\int\frac{\cos 2x}{\sqrt{\sin^2 2x + 8}} dx\]
\[\int\frac{x^2 + x - 1}{x^2 + x - 6}\text{ dx }\]
\[\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} \text{ dx }\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{1}{4 \sin^2 x + 5 \cos^2 x} \text{ dx }\]
\[\int x e^{2x} \text{ dx }\]
`int"x"^"n"."log" "x" "dx"`
` ∫ sin x log (\text{ cos x ) } dx `
\[\int\frac{\text{ log }\left( x + 2 \right)}{\left( x + 2 \right)^2} \text{ dx }\]
\[\int x^2 \tan^{- 1} x\text{ dx }\]
\[\int x \sin x \cos 2x\ dx\]
\[\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx\]
\[\int\left( x + 1 \right) \sqrt{x^2 + x + 1} \text{ dx }\]
\[\int\frac{\left( x^2 + 1 \right) \left( x^2 + 2 \right)}{\left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx\]
\[\int\frac{1}{x^4 + 3 x^2 + 1} \text{ dx }\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }\]
Write the anti-derivative of \[\left( 3\sqrt{x} + \frac{1}{\sqrt{x}} \right) .\]
\[\int\frac{\sin^2 x}{\cos^4 x} dx =\]
\[\int\frac{x^4 + x^2 - 1}{x^2 + 1} \text{ dx}\]
\[\int\frac{x^3}{\left( 1 + x^2 \right)^2} \text{ dx }\]
\[\int\frac{1}{\sqrt{3 - 2x - x^2}} \text{ dx}\]
\[\int\frac{1}{\left( \sin x - 2 \cos x \right) \left( 2 \sin x + \cos x \right)} \text{ dx }\]
\[\int\frac{1}{5 - 4 \sin x} \text{ dx }\]
\[\int\frac{1 + x^2}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{x \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}} \text{ dx}\]
