Mathematics

Solve$I= \displaystyle\int \dfrac{x+2}{x^{2}+5x+6}dx$

SOLUTION
We have,
$I=\displaystyle \int\dfrac{x+2}{x^2+5x+6}dx$
$I=\displaystyle \int\dfrac{x+2}{x^2+3x+2x+6}dx$
$I=\displaystyle \int\dfrac{x+2}{(x+3)(x+2)}dx$
$I=\displaystyle \int\dfrac{1}{(x+3)}dx$
on Integrating and we get,
$I=\log(x+3)+c \ \ \because \displaystyle \int \dfrac{1}{x}dx=\log2$

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

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Write a value of
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Q2 Single Correct Medium
Solve  $\displaystyle \int\frac{1-\sqrt{x}}{1+\sqrt{x}} dx$
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Q3 Single Correct Medium
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Q4 Assertion & Reason Hard
ASSERTION

STATEMENT-1 : $\displaystyle \int \frac{\left \{ f(x)\phi '(x)-f'(x)\phi (x) \right \}}{f(x)\phi (x)}\left \{ \log \phi (x)-\log f(x) \right \}dx=\frac{1}{2}\left \{ \log \frac{\phi(x)}{f(x)} \right \}^{2}+c$

REASON

STATEMENT-2 : $\displaystyle \int (h(x))^{n}h'(x)dx=\frac{(h(x))^{n+1}}{n+1}+c$

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is False
• C. STATEMENT-1 is False, STATEMENT-2 is True
• D. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$