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Definite Integral as Limit of Sum

If the value ...

Question

If the value of $\int \frac{d x}{cos (3 x - 2) \cdot cos (3 x + 5)}$
is $\frac{f (x)}{a sin b} + C,$ then which of the following is/are true
(where $C$ is constant of integration)

A

a = 3

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B

b = 8

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C

a + b = 10

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D

f (x) = ln ∣ ∣ ∣ \frac{sec (3 x + 5)}{sec (3 x - 2)} ∣ ∣ ∣

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Solution

The correct option is D $f (x) = ln ∣ ∣ ∣ \frac{sec (3 x + 5)}{sec (3 x - 2)} ∣ ∣ ∣$
Let $I = \int \frac{d x}{cos (3 x - 2) \cdot cos (3 x + 5)}$
$I = \int \frac{d x}{cos (3 x - 2) \cdot cos (3 x + 5)} \times \frac{sin 7}{sin 7}$
$\Rightarrow I = \frac{1}{sin 7} [\frac{sin ((3 x + 5) - (3 x - 2))}{cos (3 x - 2) \cdot cos (3 x + 5)} d x$
$\Rightarrow I = (\frac{1}{sin 7}) \int \frac{sin (3 x + 5) (cos (3 x - 2) - cos (3 x + 5) sin (3 x - 2)}{cos (3 x - 2) cos (3 x + 5)} d x$
$\Rightarrow I = \frac{1}{sin 7} \int tan (3 x + 5) - tan (3 x - 2) d x$
$\Rightarrow I = \frac{ln | sec (3 x + 5) | - ln | sec (3 x - 2) |}{3 sin 7} + C$
$∴ I = \frac{ln ∣ ∣ ∣ \frac{sec (3 x + 5)}{sec (3 x - 2)} ∣ ∣ ∣}{3 sin 7}$

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Definite Integral as Limit of Sum

Standard XII Mathematics

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