Question

Let a, b and c be positive constants. The value of ‘a’ in terms of ‘c’ if the value of integral ${\int }_0^1(ac{x}^{b+1}+a^{3}b{x}^{3b+5})$ 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

The correct option is A $\sqrt{\frac{3c}{2}}$

If the integral is independent of b, then I’(b) = 0
$\Rightarrow I^{\prime }\left(b\right)=\frac{-ac}{(b+2{)}^2}+D(\frac{a^3}{3}.\left(\frac{b+2-2}{b+2}\right))=\frac{-ac}{(b+2{)}^2}+\frac{2}{(b+2{)}^2}$
$=\frac{1}{(b+2{)}^2}(\frac{2a^3}{3}-ac)\Rightarrow \frac{2a^3}{3}=ac\Rightarrow a=\sqrt{\frac{3c}{2}}$