First slide

Evaluation of definite integrals

Question

$T h e n u m b e r s P, Q a n d R f o r w h i c h t h e f u n c t i o n$ $f (x) = P e^{2 x} + Q e^{x} + R x satisfies the conditions$ $f (0) = - 1, f^{'} (\log 2) = 31 and$ $\int_{0}^{\log 4} [f (x) - R x] d x = \frac{39}{2} a r e g i v e n b y$

Moderate

A

$P = 2, Q = - 3, R = 4$
B

$P = - 5, Q = 2, R = 3$
C

$P = 5, Q = - 2, R = 3$
D

$P = 5, Q = - 6, R = 3$

Solution

$\begin{matrix} Given f (x) = P e^{2 x} + Q e^{x} + R x \\ f (0) = - 1 and f^{'} (\log 2) = 31 \\ Hence, P + Q = - 1 \\ 2 P e^{2 \log 2} + Q e^{\log 2} + R = 31 \\ 8 P + 2 Q + R = 31 \\ \int_{0}^{\log 4} (P e^{2 x} + Q e^{x}) d x = \frac{39}{2} \\ \frac{P}{2} (e^{2 \log 4} - 1) + Q (e^{\log 4} - 1) = \frac{39}{2} \\ 15 \frac{P}{2} + 3 Q = \frac{39}{2} \\ 15 P + 6 Q = 39 \\ Solving all three equations, P = 5, Q = - 6, R = 3 \end{matrix}$

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