Question

Let $a,b$ and $c$ be positive constants. The value of ' $a$ ' in terms of ' $c$ ' if the value of integral $\underset{0}{\overset{1}{\int }}\left(ac{x}^{b+1}+a^{3}b{x}^{3b+5}\right)dx$ is independent of $b$, equals

• A. $\sqrt{\frac{3c}{2}}$
• B. $\sqrt{\frac{2c}{3}}$
• C. $\sqrt{\frac{c}{3}}$
• D. $\sqrt{\frac{3}{2c}}$

Answer 1

Answer: (A)

$I=\underset{0}{\overset{1}{\int }}\left(ac{x}^{b+1}+a^{3}b{x}^{3b+5}\right)dx={\left[ac\cdot \frac{x^{b+2}}{b+2}+\frac{a^{3}b{x}^{3b+6}}{3b+6}\right]}_0^1=\frac{ac}{b+2}+\frac{a^{3}b}{3(b+2)}$
$\text{ hence, }I(b)=\frac{1}{3(b+2)}\left[3ac+a^{3}b\right]\dots (1)\Rightarrow \frac{a^3\left(b+\frac{3c}{a^2}\right)}{3(b+2)}$
If this is independent of $b$, then $\frac{3c}{a^2}=2$