Advertisements
Advertisements
प्रश्न
\[\int\frac{x^2 + 3x - 1}{\left( x + 1 \right)^2} dx\]
योग
Advertisements
उत्तर
\[\text{Let I }= \int \left[ \frac{x^2 + 3x - 1}{\left( x + 1 \right)^2} \right]dx\]
Putting x + 1 = t
⇒ x = t – 1
& dx = dt
\[\therefore I = \int\left[ \frac{\left( t - 1 \right)^2 + 3 \left( t - 1 \right) - 1}{t^2} \right]dt\]
\[ = \int \left( \frac{t^2 - 2t + 1 + 3t - 3 - 1}{t^2} \right)dt\]
\[ = \int\left( \frac{t^2 + t - 3}{t^2} \right)dt\]
\[ = \int\left( 1 + \frac{1}{t} - 3 t^{- 2} \right)dt\]
\[ = t + \text{log} \left| t \right| - 3\left( \frac{t^{- 2 + 1}}{- 2 + 1} \right) + C\]
\[ = t + \text{log}\left| t \right| + \frac{3}{t} + C\]
\[ = x + 1 + \text{log} \left| x + 1 \right| + \frac{3}{x + 1} + C \left[ \because t = x + 1 \right]\]
Let C + 1 = C′
\[= x + \text{log} \left( x + 1 \right) + \frac{3}{x + 1} + C\prime\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\left( 3x\sqrt{x} + 4\sqrt{x} + 5 \right)dx\]
\[\int\left( 2 - 3x \right) \left( 3 + 2x \right) \left( 1 - 2x \right) dx\]
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
\[\int\frac{1}{1 - \cos 2x} dx\]
` ∫ 1/ {1+ cos 3x} ` dx
\[\int \sin^2\text{ b x dx}\]
\[\int\frac{\text{sin} \left( x - a \right)}{\text{sin}\left( x - b \right)} dx\]
` ∫ tan 2x tan 3x tan 5x dx `
\[\int \tan^{3/2} x \sec^2 \text{x dx}\]
\[\int\frac{1}{x^2 \left( x^4 + 1 \right)^{3/4}} dx\]
\[\int\frac{x^2}{\sqrt{x - 1}} dx\]
Evaluate the following integrals:
\[\int\frac{x^2}{\left( a^2 - x^2 \right)^{3/2}}dx\]
\[\int\frac{1}{\sqrt{2x - x^2}} dx\]
\[\int\frac{1}{\sqrt{5 - 4x - 2 x^2}} dx\]
\[\int\frac{e^x}{\sqrt{16 - e^{2x}}} dx\]
\[\int\frac{\cos x}{\sqrt{\sin^2 x - 2 \sin x - 3}} dx\]
\[\int\frac{x^2 \left( x^4 + 4 \right)}{x^2 + 4} \text{ dx }\]
\[\int\frac{x + 2}{\sqrt{x^2 - 1}} \text{ dx }\]
`int 1/(sin x - sqrt3 cos x) dx`
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int e^x \left( \frac{\sin 4x - 4}{1 - \cos 4x} \right) dx\]
∴\[\int e^{2x} \left( - \sin x + 2 \cos x \right) dx\]
\[\int\left( \frac{1}{\log x} - \frac{1}{\left( \log x \right)^2} \right) dx\]
\[\int\sqrt{2ax - x^2} \text{ dx}\]
\[\int(2x + 5)\sqrt{10 - 4x - 3 x^2}dx\]
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
\[\int\frac{1}{\left( x + 1 \right)^2 \left( x^2 + 1 \right)} dx\]
Find \[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)^2}dx\]
\[\int\frac{x^2 - 3x + 1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{\cos 2x - 1}{\cos 2x + 1} dx =\]
\[\int\sin x \sin 2x \text{ sin 3x dx }\]
\[\int\text{ cos x cos 2x cos 3x dx}\]
\[\int\frac{\sin x}{\sqrt{1 + \sin x}} dx\]
\[\int \sin^5 x\ dx\]
\[\int\frac{5x + 7}{\sqrt{\left( x - 5 \right) \left( x - 4 \right)}} \text{ dx }\]
\[\int\frac{1}{\left( \sin x - 2 \cos x \right) \left( 2 \sin x + \cos x \right)} \text{ dx }\]
\[\int\sqrt{a^2 + x^2} \text{ dx }\]
\[\int \left( \sin^{- 1} x \right)^3 dx\]
