Mathematics

# Evaluate$\displaystyle\int { \dfrac { { x }^{ 3 }+{ 4x }^{ 2 }-7x+5 }{ x+2 } } dx$

##### SOLUTION
$\displaystyle\int \dfrac{x^3+4x^2-7x+5}{x+2}dx$

$x^2+2x-11$
________________
$x+2)x^3+4x^2-7x+5$
$x^3+2x^2$
________________
$2x^2-7x$
$2x^2-4x$
______________
$-11x+5$
$-11x-22$
____________
$27$
____________

So $\displaystyle\int x^2+2x-11+\dfrac{27}{x+2}dx$

$\Rightarrow\dfrac{x^3}{3}+\dfrac{2x^2}{2}-11x+27 log(x+2)+c$

$\Rightarrow\dfrac{x^3}{3}+x^2-11x+27 log(x+2)+c$.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
$\displaystyle \int { \left\{ \log { \left( \log { x } \right) +\dfrac { 1 }{ { \left( \log { x } \right) }^{ 2 } } } \right\} } dx$

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Q2 Single Correct Medium
$\displaystyle \int{\dfrac{1}{x+\sqrt{x-1}}dx}$=
• A. $\log(x+\sqrt{x-1})+c$
• B. $\log(x+\sqrt{x-1})+-\dfrac{2}{\sqrt{3}}.\tan^{-1}\left( \dfrac{2\sqrt{x-1}+1}{\sqrt{3}}\right)+c$
• C. $None\ of\ these$
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Q3 Subjective Medium
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Q4 Single Correct Hard
If $\displaystyle\int{\frac{dx}{(x^2+1)(x^2+4)}}=k\tan^{-1}{x}+l\tan^{-1}{\frac{x}{2}}+C$, then
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Q5 Assertion & Reason Hard
##### ASSERTION

Let f : R $\rightarrow$ R be defined as $f(x) = ax^2 + bx + c$, where a, b, c $\varepsilon$ R and a $\neq$ 0.

If $f(x) = 0$ is having non-real roots, then $\displaystyle \int \frac{dx}{f(x)} = \lambda tan^{-1} (g(x)) + k$, where $\lambda$, $k$ are constants and $g(x)$ is linear function of $x$.

##### REASON

$tan(tan^{-1} g(x)) = g(x) \forall x \in R$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion