Question

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

• A. $\frac{x^4}{4}+\frac{x^2}{2}+x+c$
• B. $\frac{x^4}{4}+\frac{x^2}{3}+x^2+2x+c$
• C. $\frac{x^4}{4}+\frac{x^3}{3}+\frac{x^2}{2}+x+c$
• D. $\frac{x^4}{4}+\frac{x^3}{3}+c$

Answer 1

Answer: (C)

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

Consider $6\int f\left(x\right)g\left(x\right)dx=x^6+3x^4+3x^2+c$

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

Differentiate w.r.t $x$,

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

Consider, $2\int \frac{g\left(x\right)dx}{f\left(x\right)}=x^2+c$

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

Differentiate w.r.t $x$

$\Rightarrow \frac{g\left(x\right)}{f\left(x\right)}=x$ ............ $\left(2\right)$

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

$\frac{\left(1\right)}{\left(2\right)}\Rightarrow \left(f\right(x){)}^2=x^4+2x^2+1=(x^2+1{)}^2$
$\Rightarrow f\left(x\right)=x^2+1$

Then, $\int \left(g\right(x)+f(x\left)\right)dx=\int (x^3+x+x^2+1)dx$
$=\frac{x^4}{4}+\frac{x^3}{3}+\frac{x^2}{2}+x+c$