Mathematics

# Evaluate the following $\displaystyle \underset{1}{\overset{2}{\int}} \dfrac{5x^2}{x^2 + 4x + 3}$

##### SOLUTION
$\int_{1}^{2}\dfrac{5x^2}{x^2+4x+3}dx$

$=5\int_{1}^{2}1+\dfrac{-4x-3}{x^2+4x+3}dx$

$=5\int_{1}^{2}1+\dfrac{-4x-3}{(x+2)^2-1}dx$

$u=x+2\rightarrow du=dx$

$=5\int_{1}^{2}1+\dfrac{-4u+5}{u^2-1}du$

$=5\left ( u-2\ln \left | u^2-1 \right |-5\left ( \frac{1}{2}\ln \left | u+1 \right |-\frac{1}{2}\ln \left | u-1 \right | \right ) \right )$

$=5\left ( x+2-2\ln \left | (x+2)^2-1 \right |-5\left ( \frac{1}{2}\ln \left | x+2+1 \right |-\frac{1}{2}\ln \left | x+2-1 \right | \right ) \right )$

$=5\left ( x+2-2\ln \left | x^2+4x+3\right |-5\left ( \frac{1}{2}\ln \left | x+3 \right |-\frac{1}{2}\ln \left | x+1 \right | \right ) \right )+C$

$\int_{1}^{2}\dfrac{5x^2}{x^2+4x+3}dx=5\left ( 4-2\ln (15)-5\frac{\ln 5 -\ln 3}{2}\right )-5\left ( 3+5\dfrac{\ln2}{2}-11\ln 2 \right )$

$=5+55\ln 2-10\ln 15+\dfrac{-25\ln\frac{5}{3}-25\ln 2}{2}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 One Word Medium
$\displaystyle \int \frac{e^{2x}dx}{\sqrt[4]{\left ( e^{x}+1 \right )}}=\frac{k}{21}\left ( e^{x}+1 \right )^{3/4}\left [ 3e^{x}-4 \right ].$ Find the value of $k$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
If $f$ is continuous for any $t\epsilon R$
then for  $I_{1}=\displaystyle \int_{\sin ^{2}t}^{1+\cos ^{2}t}xf\left ( x \left ( 2-x \right )\right )dx$ and $I_{2}= \displaystyle \int_{\sin ^{2}t}^{1+\cos ^{2}t}f\left ( x \left ( 2-x \right )\right )dx$
which of the following is not true?
• A. $I_{1}= \displaystyle \int_{\sin ^{2}t}^{1+\cos ^{2}t}f\left ( x\left ( 2-x \right ) \right )dx$
• B. $I_{1}+I_{2}= 2I_{2}$
• C. $I_{1}+I_{2}= 2I_{1}$
• D. None of the above

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
Evaluate the integral, $\int _{ 0 }^{ 1 }{ \cos { \left( 2\cot ^{ -1 }{ \sqrt { \dfrac { 1-x }{ 1+x } } } \right) } } dx=$
• A. $1/2$
• B. $0$
• C. $1$
• D. $-1/2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle I = \int_{0}^{\infty}\frac{\sqrt{x}\:d\:x}{(1+x)(2+x)(3+x)}$ then $I$ equals
• A. $\displaystyle \frac{\pi}{2}(2\sqrt{2}+\sqrt{3}-1)$
• B. $\displaystyle \frac{\pi}{2}(2\sqrt{2}-\sqrt{3}+1)$
• C. None of these
• D. $\displaystyle \frac{\pi}{2}(2\sqrt{2}-\sqrt{3}-1)$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$