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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 $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :
Mathematics

$$\displaystyle \int \left ( f(x)+g(x)\right)dx $$ is equal to


Answer & Solution

$$\displaystyle \frac {x^{4}}{4}+\frac {x^{3}}{3}+\frac {x^{2}}{2}+x+c$$


SOLUTION
Consider $$\displaystyle 6\int  f(x)g(x)dx=x^{ 6 }+3x^{ 4 }+3x^{ 2 }+c$$

$$\Rightarrow \displaystyle \int  f(x)g(x)dx=\dfrac { 1 }{ 6 } (x^{ 6 }+3x^{ 4 }+3x^{ 2 }+c)$$

 Differentiate w.r.t $$x$$, 

 $$\Rightarrow f(x)g(x)=x^{ 5 }+2x^{ 3 }+x$$ ........... $$(1)$$

Consider, $$\displaystyle 2\int \dfrac { g(x)dx }{ f(x) } =x^{ 2 }+c$$

$$\Rightarrow \displaystyle \int  \dfrac { g(x)dx }{ f(x) } =\frac { 1 }{ 2 } (x^{ 2 }+c)$$

Differentiate w.r.t $$x$$

$$\Rightarrow \dfrac { g(x) }{ f(x) } =x$$ ............ $$(2)$$

$$(1)\times (2)\Rightarrow ({ g(x)) }^{ 2 }=x^{ 6 }+2x^{ 4 }+x^{ 2 }=({ x }^{ 3 }+x)^{ 2 }$$
 $$\Rightarrow g(x)={ x }^{ 3 }+x$$

 $$\dfrac { (1) }{ (2) } \Rightarrow (f(x))^{ 2 }=x^{ 4 }+2x^{ 2 }+1=({ x }^{ 2 }+1)^{ 2 }$$
 $$\Rightarrow f(x)={ x }^{ 2 }+1$$

 Then, $$\displaystyle \int { (g(x)+f(x))dx } =\int (x^{3}+x+{x}^{2}+1)dx$$
 $$=\dfrac { { x }^{ 4 } }{ 4 } +\dfrac { { x }^{ 3 } }{ 3 } +\dfrac { x^{ 2 } }{ 2 } +x+c$$
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Single Correct Hard Published on 17th 08, 2020
Mathematics

$$\displaystyle f'(1) + g'(2)$$ is equal to


Answer & Solution

$$15$$


SOLUTION
Consider $$\displaystyle 6\int  f(x)g(x)dx=x^{ 6 }+3x^{ 4 }+3x^{ 2 }+c$$
$$\Rightarrow \displaystyle \int  f(x)g(x)dx=\dfrac { 1 }{ 6 } (x^{ 6 }+3x^{ 4 }+3x^{ 2 }+c)$$

 Differentiate w.r.t $$x$$, 

 $$\Rightarrow f(x)g(x)=x^{ 5 }+2x^{ 3 }+x$$ ........... $$(1)$$

Consider, $$\displaystyle 2\int \dfrac { g(x)dx }{ f(x) } =x^{ 2 }+c$$

$$\Rightarrow \displaystyle \int  \dfrac { g(x)dx }{ f(x) } =\frac { 1 }{ 2 } (x^{ 2 }+c)$$

Differentiate w.r.t $$x$$

$$\Rightarrow \dfrac { g(x) }{ f(x) } =x$$ ............ $$(2)$$

$$(1)\times (2)\Rightarrow ({ g(x)) }^{ 2 }=x^{ 6 }+2x^{ 4 }+x^{ 2 }=({ x }^{ 3 }+x)^{ 2 }$$
 $$\Rightarrow g(x)={ x }^{ 3 }+x$$

 $$\dfrac { (1) }{ (2) } \Rightarrow (f(x))^{ 2 }=x^{ 4 }+2x^{ 2 }+1=({ x }^{ 2 }+1)^{ 2 }$$
 $$\Rightarrow f(x)={ x }^{ 2 }+1$$

Now, $$f'(x)=2x$$ and $$g'(x)=3x^2+1$$
$$\Rightarrow f'(1)+g'(2)=2+13=15$$
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Single Correct Medium Published on 17th 08, 2020
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