First slide

Evaluation of definite integrals

Question

The numbers P, Q and R for which the function $f (x) = P e^{2 x} + Q e^{x} + R x$ satisfies $f (0) = - 1$
$f^{'} (\log 2) = 31$ and $\int_{0}^{\log 4} [f (x) - R x] d x = 39 / 2$ are given by

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

We have $f^{'} (x) = 2 P e^{2 x} + Q e^{x} + R$ , so that $31 = f^{'} (\log 2) = 8 P + 2 Q + R$
Also, $- 1 = f (0) = P + Q$ . Further,
$\begin{array}{l} \frac{39}{2} = \int_{0}^{\log 4} [f (x) - R x] d x = \int_{0}^{\log 4} [P e^{2 x} + Q e^{x}] d x \\ = \frac{P}{2} e^{2 x} + {Q e^{x}|}_{0}^{\log 4} = \frac{P}{2} \times 16 + 4 Q - \frac{P}{2} - Q \\ = \frac{15 P}{2} + 3 Q \end{array}$
Solving the above equations, we get $P = 5, Q = - 6$ and $R = 3$

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