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∫ex(1+x2)(1+x)2 dx - Mathematics and Statistics

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प्रश्न

$\int e^{x} \frac{(1 + x^{2})}{{(1 + x)}^{2}} d x$

योग

उत्तर

Let I = $\int e^{x} \frac{(1 + x^{2})}{{(1 + x)}^{2}} d x$

= $\int e^{x} [\frac{x^{2} - 1 + 2}{{(1 + x)}^{2}}] d x$

= $\int e^{x} [\frac{x^{2} - 1}{{(x + 1)}^{2}} + \frac{2}{{(x + 1)}^{2}}] d x$

= $\int e^{x} [\frac{x -}{x + 1} + \frac{2}{{(x + 1)}^{2}}] d x$

Put f(x) = $\frac{x - 1}{x + 1}$

∴ f'(x) = $\frac{(x + 1) (1 - 0) - (x - 1) (1 + 0)}{{(x + 1)}^{2}}$

= $\frac{2}{{(x + 1)}^{2}}$

∴ I = $\int e^{x} [f (x) + f' (x)] d x$

= e^x.f(x) + c

= $e^{x} (\frac{x - 1}{x + 1}) + c$

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Methods of Integration: Integration Using Partial Fractions

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

अध्याय 2.3: Indefinite Integration - Short Answers II

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एससीईआरटी महाराष्ट्र Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

अध्याय 2.3 Indefinite Integration
Short Answers II | Q 13

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Methods of Integration: Integration Using Partial Fractions

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