The integral 0 4 136 sin x 3 sin x + 5 cos x ? d x is equal to: [2024]

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Solution

Q.
$The integral \int_{0}^{\frac{π}{4}} \frac{136 \sin x}{3 \sin x + 5 \cos x} d x is equal to:$ [2024]

1 $3 π - 30 \log_{e} 2 + 20 \log_{e} 5$

2 $3 π - 50 \log_{e} 2 + 20 \log_{e} 5$

3 $3 π - 25 \log_{e} 2 + 10 \log_{e} 5$

4 $3 π - 10 \log_{e} (2 \sqrt{2}) + 10 \log_{e} 5$

Ans.
(2)

Let $I = \int_{0}^{π / 4} \frac{136 \sin x}{3 \sin x + 5 \cos x} d x$

Let $\sin x = λ (3 \sin x + 5 \cos x) + μ (3 \cos x - 5 \sin x)$

For $x = 0,$ we have $0 = 5 λ + 3 μ$

and for $x = π / 2,$ we have $1 = 3 λ - 5 μ$

Solving these two equations, we get $λ = \frac{3}{34}$ and $μ = \frac{- 5}{34}$

$∴$ $I = 136 \int_{0}^{π / 4} \frac{3}{34} (\frac{3 \sin x + 5 \cos x}{3 \sin x + 5 \cos x}) d x - \frac{136 \times 5}{34} \int_{0}^{π / 4} \frac{3 \cos x - 5 \sin x}{3 \sin x + 5 \cos x} d x$

$= \frac{136 \times 3}{34} {[x]}_{0}^{π / 4} - \frac{136 \times 5}{34} \ln | 3 \sin x + 5 \cos x |_{0}^{π / 4}$

$= 3 π - 20 [\ln \frac{8}{\sqrt{2}} - \ln 5] = 3 π - 20 \ln 4 \sqrt{2} + 20 \ln 5$

$= 3 π - 20 \ln {(2)}^{\frac{5}{2}} + 20 \ln 5 = 3 π - 50 \ln 2 + 20 \ln 5$

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