Mathematics

# Value of $\int \:e^x\:\frac{\left(x^2+3x+3\right)}{\left(x+2\right)^2}dx\:$ is

$e^x\left[\dfrac{x+1}{x+2}\right]+c$

##### SOLUTION
$\int{{e}^{x}\left(\dfrac{{x}^{2}+3x+3}{{\left(x+2\right)}^{2}}\right)dx}$
$=\int{{e}^{x}\left(1-\dfrac{x+1}{{\left(x+2\right)}^{2}}\right)dx}$
$=\int{{e}^{x}dx}-\int{{e}^{x}\left(\dfrac{x+1}{{\left(x+2\right)}^{2}}\right)dx}$
$=\int{{e}^{x}dx}-\int{{e}^{x}\left(\dfrac{x+2-1}{{\left(x+2\right)}^{2}}\right)dx}$ take $1=2-1$
$=\int{{e}^{x}dx}-\int{{e}^{x}\left(\dfrac{x+2}{{\left(x+2\right)}^{2}}-\dfrac{1}{{\left(x+2\right)}^{2}}\right)dx}$
$=\int{{e}^{x}dx}-\int{{e}^{x}\left(\dfrac{1}{x+2}-\dfrac{1}{{\left(x+2\right)}^{2}}\right)dx}$
Let $f\left(x\right)=\dfrac{1}{x+2}$ then ${f}^{\prime}{\left(x\right)}=\dfrac{-1}{{\left(x+2\right)}^{2}}$
We have $\int{{e}^{x}\left(f\left(x\right)+{f}^{\prime}\left(x\right)\right)}={e}^{x}f\left(x\right)+c$
Using this we have $\int{{e}^{x}\left(\dfrac{1}{x+2}-\dfrac{1}{{\left(x+2\right)}^{2}}\right)dx}={e}^{x}\left(\dfrac{1}{x+2}\right)+c$
$\therefore \int{{e}^{x}\left(\dfrac{{x}^{2}+3x+3}{{\left(x+2\right)}^{2}}\right)dx}$
$=\int{{e}^{x}dx}-\int{{e}^{x}\left(\dfrac{1}{x+2}-\dfrac{1}{{\left(x+2\right)}^{2}}\right)dx}$
$={e}^{x}-{e}^{x}\left(\dfrac{1}{x+2}\right)+c$
$={e}^{x}\left(1-\left(\dfrac{1}{x+2}\right)\right)+c$
$={e}^{x}\left(\dfrac{x+2-1}{x+2}\right)+c$
$={e}^{x}\left(\dfrac{x+1}{x+2}\right)+c$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

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Q5 Passage Hard

In calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

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Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration  by parts is especially useful when calculating $\int P_{n}(x)\, Q(x)\, dx$, where $P_{n}(x)$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.