Mathematics

# Let $F(x)$ be the primitive of $\displaystyle \frac{3x+2}{\sqrt{x-9}}$ w.r.t $x$ . If $F(10)=60$ then the value of $F(13)$, is

$132$

##### SOLUTION

$F(x)=\displaystyle \int \frac{3x+2}{\sqrt{x-9}}dx$
Let, $\displaystyle \sqrt{x-9}=t \Rightarrow \frac{1}{2\sqrt{x-9}}=\frac{\mathrm{d} t}{\mathrm{d} x}$
$x-9=t^{2} \Rightarrow x=t^{2}+9$
So, $\displaystyle \int \frac{3x+2}{\sqrt{x-9}}dx=\int 2(3t^{2}+29)dt=2(t^{3}+29t)+c$
$F(10)=60, t=1,$
$2(1+29)+c=60 \Rightarrow c=0$
$F(13) \Rightarrow t=2,$
$=2(8+58)=2(66)=132$

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

#### Realted Questions

Q1 Single Correct Medium
If $\int \dfrac{x \tan^{-1} x}{\sqrt{1+x^{2}}} dx=\sqrt{1+x^{2}}\ f(x)+A\ \ell n |x+\sqrt{x^{2}+1}|+C$, then
• A. $f(x)=\tan^{-1}x, A=1$
• B. $f(x)=2 \tan^{-1} x, A=-1$
• C. $f(x)=2 \tan^{-1}x, A=1$
• D. $f(x)=\tan^{-1}x, A=-1$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\displaystyle \int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \dfrac { dx }{ \left[ x \right] +\left[ sinx \right] +4 } }$, where $[t]$ denotes the greatest integer less than or equal to $t$, is :
• A. $\dfrac { 3 }{ 10 } \left( 4\pi -3 \right)$
• B. $\dfrac { 1 }{ 12 } \left( 7\pi -5 \right)$
• C. $\dfrac { 3 }{ 20 } \left( 4\pi -3 \right)$
• D. $\dfrac { 1 }{ 12 } \left( 7\pi +5 \right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle \int \dfrac{\text{cosec}^2x - 2010}{\cos^{2010x}}dx=-\dfrac{f(x)}{(g(x))^{2010}}+C$; then the number of solutions where equation $\dfrac{f(x)}{g(x)}=\left \{ x \right \}$ in $[0, 2\pi]$ is/are:
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1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Find solution in terms of indefinite integration, using substitution
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