If - ex 1-x1-x2 2dx-exca-bx2-k - then the sum of a-b and c is -

∫ e^x ( 1 x1+x^2 )^2dx = ∫ e^x(1 2x+x^2)(1+x^2)^2dx=∫ e^x ( 11+x^2 2x(1+x^2)^2 )dx =e^x1+x^2+k Using the property ∫ e^x(f(x)+f'(x))=e^xf(x) ∴ a=1,b=1,c=1 ...

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Engineering Mathematics

Review of Integration II ( Rational functions and Universal Substitutions)

If ∫ ex 1-x1+...

Question

$If \int e^{x} {(\frac{1 - x}{1 + x^{2}})}^{2} d x = \frac{e^{x c}}{a + b x^{2}} + k, then the sum of a,b and c is ___.$

3

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Solution

The correct option is A 3
$\int e^{x} {(\frac{1 - x}{1 + x^{2}})}^{2} d x = \int e^{x} \frac{(1 - 2 x + x^{2})}{(1 + x^{2})^{2}} d x = \int e^{x} (\frac{1}{1 + x^{2}} - \frac{2 x}{(1 + x^{2})^{2}}) d x$

$= \frac{e^{x}}{1 + x^{2}} + k$

Using the property

$\int e^{x} (f (x) + f^{'} (x)) = e^{x} f (x)$

$∴ a = 1, b = 1, c = 1$

sum $= a + b + c = 3$

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Similar questions

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

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

\int e^{x} {(\frac{1 - x}{1 + x^{2}})}^{2} d x is equal to (a) \frac{e^{x}}{1 + x} + C (b) - \frac{e^{x}}{1 + x^{2}} + C (c) \frac{e^{x}}{(1 + x}

²)2+C (d) -ex(1+x²)2+C

Q. If

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

, then I =

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

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If ∫ex(1−x1+x2)2dx=exca+bx2+k, then the sum of a,b and c is ___. 3

The correct option is A 3∫ex(1−x1+x2)2dx=∫ex(1−2x+x2)(1+x2)2dx=∫ex(11+x2−2x(1+x2)2)dx =ex1+x2+k Using the property ∫ex(f(x)+f′(x))=exf(x) ∴ a=1,b=1,c=1 sum =a+b+c=3

$If \int e^{x} {(\frac{1 - x}{1 + x^{2}})}^{2} d x = \frac{e^{x c}}{a + b x^{2}} + k, then the sum of a,b and c is ___.$

3