Let a, b and c be positive constants. The val | vedclass

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(A) Let $I = \int_{0}^{1} (acx^{b+1} + a^3bx^{3b+5}) \, dx$. Evaluating the integral: $I = \left[ \frac{acx^{b+2}}{b+2} + \frac{a^3bx^{3b+6}}{3b+6} \r

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$\sqrt{\frac{3c}{2}}$

Vedclass · Answer

(A) Let $I = \int_{0}^{1} (acx^{b+1} + a^3bx^{3b+5}) \, dx$. Evaluating the integral: $I = \left[ \frac{acx^{b+2}}{b+2} + \frac{a^3bx^{3b+6}}{3b+6} \right]_{0}^{1}$ $I = \frac{ac}{b+2} + \frac{a^3b}{3(b+2)}$ $I = \frac{1}{3(b+2)} [3ac + a^3b]$ $I = \frac{a^3}{3} \left( \frac{b + \frac{3c}{a^2}}{b+2} \right)$ For the integral to be independent of $b$,the expression must be a constant. This happens when the ratio of the coefficients of $b$ in the numerator and denominator is equal to the ratio of the constant terms. Thus,$\frac{1}{1} = \frac{3c/a^2}{2}$ $2 = \frac{3c}{a^2}$ $a^2 = \frac{3c}{2}$ $a = \sqrt{\frac{3c}{2}}$.

Let $a, b$ and $c$ be positive constants. The value of $a$ in terms of $c$ if the value of the integral $\int_{0}^{1} (acx^{b+1} + a^3bx^{3b+5}) \, dx$ is independent of $b$ is:

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