Advertisements
Advertisements
प्रश्न
\[\int\left\{ \sqrt{x}\left( a x^2 + bx + c \right) \right\} dx\]
योग
उत्तर
\[\int\sqrt{x} \left( a x^2 + bx + c \right)dx\]
\[ = \int x^\frac{1}{2} \left( a x^2 + bx + c \right)dx\]
`=∫ (ax^{2 + 1/2} + bx^{1/2+1 }+ c x^{1/2})dx`
`= a∫ x^{5/2 }dx + b∫ x^{3/2 }dx + c∫ x^{1/2 } dx`
`= a [ x^(5/2 + 1)/(5/2+ 1) ]+ b[ x^(3/2+1)/(3/2+ 1) ] + c[ x^(1/2+1)/(1/2 + 1 )]+ C`
\[ = \frac{2a}{7} x^\frac{7}{2} + \frac{2b}{5} x^\frac{3}{2} + \frac{2c}{3} x^\frac{3}{2} + C\]
\[ = \int x^\frac{1}{2} \left( a x^2 + bx + c \right)dx\]
`=∫ (ax^{2 + 1/2} + bx^{1/2+1 }+ c x^{1/2})dx`
`= a∫ x^{5/2 }dx + b∫ x^{3/2 }dx + c∫ x^{1/2 } dx`
`= a [ x^(5/2 + 1)/(5/2+ 1) ]+ b[ x^(3/2+1)/(3/2+ 1) ] + c[ x^(1/2+1)/(1/2 + 1 )]+ C`
\[ = \frac{2a}{7} x^\frac{7}{2} + \frac{2b}{5} x^\frac{3}{2} + \frac{2c}{3} x^\frac{3}{2} + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\left( x^e + e^x + e^e \right) dx\]
\[\int\frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} dx\]
\[\int\frac{x^2 + x + 5}{3x + 2} dx\]
\[\int\frac{x^2 + 3x - 1}{\left( x + 1 \right)^2} dx\]
\[\int\frac{1}{\sqrt{1 + \cos x}} dx\]
\[\int\frac{e^x + 1}{e^x + x} dx\]
\[\int\frac{x^2}{\sqrt{x - 1}} dx\]
\[\int\frac{x^2 + 3x + 1}{\left( x + 1 \right)^2} dx\]
\[\int\frac{x^2}{\sqrt{1 - x}} dx\]
` ∫ tan^3 x sec^2 x dx `
\[\int \sin^5 x \text{ dx }\]
\[\int\frac{1}{4 x^2 + 12x + 5} dx\]
\[\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)\]
\[\int\frac{\sin 2x}{\sqrt{\sin^4 x + 4 \sin^2 x - 2}} dx\]
` ∫ {x-3} /{ x^2 + 2x - 4 } dx `
\[\int\frac{x + 2}{2 x^2 + 6x + 5}\text{ dx }\]
\[\int\frac{6x - 5}{\sqrt{3 x^2 - 5x + 1}} \text{ dx }\]
\[\int\frac{1}{\sqrt{3} \sin x + \cos x} dx\]
\[\int x \text{ sin 2x dx }\]
\[\int \tan^{- 1} \left( \sqrt{x} \right) \text{dx }\]
\[\int\left( \tan^{- 1} x^2 \right) x\ dx\]
\[\int e^x \cdot \frac{\sqrt{1 - x^2} \sin^{- 1} x + 1}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{x^2 + 1}{x^2 - 1} dx\]
\[\int\frac{1}{\sin x + \sin 2x} dx\]
\[\int\sqrt{\cot \text{θ} d } \text{ θ}\]
\[\int x^{\sin x} \left( \frac{\sin x}{x} + \cos x . \log x \right) dx\] is equal to
If \[\int\frac{\cos 8x + 1}{\tan 2x - \cot 2x} dx\]
\[\int\frac{x^9}{\left( 4 x^2 + 1 \right)^6}dx\] is equal to
\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x}\]
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} \text{ dx }\]
\[\int\frac{1}{1 - 2 \sin x} \text{ dx }\]
\[\int\frac{1}{\sin^4 x + \cos^4 x} \text{ dx}\]
\[\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} \text{ dx }\]
\[\int\left( 2x + 3 \right) \sqrt{4 x^2 + 5x + 6} \text{ dx}\]
\[\int\frac{\log \left( 1 - x \right)}{x^2} \text{ dx}\]
\[\int x^3 \left( \log x \right)^2\text{ dx }\]
\[\int\frac{x^2}{\sqrt{1 - x}} \text{ dx }\]
\[\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }\]
\[\int\frac{x^2 - 2}{x^5 - x} \text{ dx}\]
\[\int\frac{\cot x + \cot^3 x}{1 + \cot^3 x} \text{ dx}\]
