Question

$\text{If}\int \frac{x^3+x^2+x}{\sqrt{15+12x+10x^2}}dx=\frac{g\left(x\right)}{k}.\sqrt{15+12x+10x^2}+C,$ where $C$ is arbitrary constant of integration and $g\left(1\right)=1$, then $\left(g^{\prime }\right(1)+g"(1)+k)$ is

• A. 30
• B. 32
• C. 34
• D. 36

Answer 1

The correct option is C 34

Multiply $N^r\&D^r$ by $x^2$
$\int \frac{x^5+x^4+x^3}{\sqrt{10x^6+12x^5+15x^4}}dx$
$10x^6+12x^5+15x^4=t^2$
$\int \frac{dt}{30}=\frac{t}{30}+C=\frac{x^2}{30}\sqrt{15+12x+10x^2}+C$
$\Rightarrow k=30\text{and}g\left(x\right)=x^2\Rightarrow g^{\prime }\left(1\right)=2\text{and}{g}^{\prime \prime }\left(1\right)=2\Rightarrow \left(g^{\prime }\right(1)+g"(1)+k)=2+2+30=34$