Question

Let $a, b$ and $c$ be positive constants. The value of ' $a$ ' in terms of ' $c$ ' if the value of integral $\int\limits_0^1\left(a c x^{b+1}+a^3 b x^{3 b+5}\right) d x$ is independent of $b$, equals

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

Answer 1

Answer: (A)

$ I=\int\limits_0^1\left(a c x^{b+1}+a^3 b x^{3 b+5}\right) d x=\left[a c \cdot \frac{x^{b+2}}{b+2}+\frac{a^3 b x^{3 b+6}}{3 b+6}\right]_0^1=\frac{a c}{b+2}+\frac{a^3 b}{3(b+2)} $
$\text { hence, } I(b)=\frac{1}{3(b+2)}\left[3 a c+a^3 b\right] \ldots(1) \Rightarrow \frac{a^3\left(b+\frac{3 c}{a^2}\right)}{3(b+2)} $
If this is independent of $b$, then $\frac{3 c}{a^2}=2$