Advertisements
Advertisements
प्रश्न
\[\int\frac{1}{4 x^2 + 4x + 5} dx\]
योग
उत्तर
\[\text{ Let I }= \int\frac{dx}{4 x^2 + 4x + 1 + 4}\]
\[ = \int\frac{dx}{\left( 2x \right)^2 + 2 \times 2x + 1 + 22}\]
\[ = \int\frac{dx}{\left( 2x + 1 \right)^2 + 2^2}\]
\[\text{ Putting }\left( 2x + 1 \right) = t\]
\[ \Rightarrow 2 \text{ dx = dt }\]
\[ \Rightarrow dx = \frac{dt}{2}\]
\[ \therefore I = \frac{1}{2}\int\frac{dt}{t^2 + 2^2}\]
\[ = \frac{1}{2} \times \frac{1}{2} \text{ tan}^{- 1} \left( \frac{t}{2} \right) + C\]
\[ = \frac{1}{4} \text{ tan}^{- 1} \left( \frac{2x + 1}{2} \right) + C ....................\left[ \because t = \left( 2x + 1 \right) \right]\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
If f' (x) = 8x3 − 2x, f(2) = 8, find f(x)
\[\int\frac{1}{2 - 3x} + \frac{1}{\sqrt{3x - 2}} dx\]
\[\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx\]
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
\[\int\frac{1}{x (3 + \log x)} dx\]
\[\int\frac{x + 1}{x \left( x + \log x \right)} dx\]
\[\int\frac{\log\left( 1 + \frac{1}{x} \right)}{x \left( 1 + x \right)} dx\]
\[\int x^3 \cos x^4 dx\]
\[\int\frac{\cos\sqrt{x}}{\sqrt{x}} dx\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
` ∫ tan^5 x dx `
\[\int \cot^6 x \text{ dx }\]
\[\int \cos^5 x \text{ dx }\]
` ∫ {1}/{a^2 x^2- b^2}dx`
\[\int\frac{1}{\sqrt{1 + 4 x^2}} dx\]
\[\int\frac{1}{\sqrt{3 x^2 + 5x + 7}} dx\]
\[\int\frac{\sin x}{\sqrt{4 \cos^2 x - 1}} dx\]
` ∫ {x-3} /{ x^2 + 2x - 4 } dx `
\[\int\frac{x - 1}{3 x^2 - 4x + 3} dx\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
\[\int\frac{1}{1 - \sin x + \cos x} \text{ dx }\]
\[\int\frac{1}{5 + 7 \cos x + \sin x} dx\]
\[\int\frac{8 \cot x + 1}{3 \cot x + 2} \text{ dx }\]
\[\int\frac{\sin^{- 1} x}{x^2} \text{ dx }\]
\[\int x \cos^3 x\ dx\]
\[\int e^x \frac{x - 1}{\left( x + 1 \right)^3} \text{ dx }\]
\[\int\left( 2x - 5 \right) \sqrt{2 + 3x - x^2} \text{ dx }\]
\[\int\left( 2x + 3 \right) \sqrt{x^2 + 4x + 3} \text{ dx }\]
\[\int\frac{5x}{\left( x + 1 \right) \left( x^2 - 4 \right)} dx\]
\[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 3 \right)} dx\]
\[\int\frac{dx}{\left( x^2 + 1 \right) \left( x^2 + 4 \right)}\]
\[\int e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx\]
\[\int \sin^4 2x\ dx\]
\[\int\sin x \sin 2x \text{ sin 3x dx }\]
\[\int\frac{\sin x}{\cos 2x} \text{ dx }\]
\[\int\sqrt{1 + 2x - 3 x^2}\text{ dx } \]
\[\int \tan^{- 1} \sqrt{x}\ dx\]
