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JEE Main 20245 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Definite Integration

The integral ∫_0^ π / 4 (136 x)/(3 x+5 x) d x is equal to :

The integral ∫0π/4136sin⁡x3sin⁡x+5cos⁡xdx\int_0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} d x is equal to :

Options

  1. A3π−50log⁡e2+20log⁡e53 \pi-50 \log _e 2+20 \log _e 5
  2. B3π−25log⁡e2+10log⁡e53 \pi-25 \log _e 2+10 \log _e 5
  3. C3π−10log⁡e(22)+10log⁡e53 \pi-10 \log _e(2 \sqrt{2})+10 \log _e 5
  4. D3π−30log⁡e2+20log⁡e53 \pi-30 \log _e 2+20 \log _e 5

Correct answer

A. 3π−50log⁡e2+20log⁡e53 \pi-50 \log _e 2+20 \log _e 5

Step-by-step solution

I=∫0π/4136sin⁡x3sin⁡x+5cos⁡xdx136sin⁡x=A(3sin⁡x+5cos⁡x)+B(3cos⁡x−5sin⁡x)136=3A−5B...(i)0=5A+3B...(ii)\begin{aligned} & I=\int_0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} d x \\ & 136 \sin x=A(3 \sin x+5 \cos x)+B(3 \cos x-5 \sin x) \\ & 136=3 A-5 B...(i) \\ & 0=5 A+3 B...(ii)\end{aligned}3 B=−5 A⇒B=−53 A136=3 A−5(−53 A)136=3 A+253 A136=34 A3⇒A=136×334=12 B=−53(12)=−20\begin{aligned} & 3 \mathrm{~B}=-5 \mathrm{~A} \Rightarrow \mathrm{B}=-\frac{5}{3} \mathrm{~A} \\ & 136=3 \mathrm{~A}-5\left(-\frac{5}{3} \mathrm{~A}\right) \\ & 136=3 \mathrm{~A}+\frac{25}{3} \mathrm{~A} \\ & 136=\frac{34 \mathrm{~A}}{3} \\ & \Rightarrow \mathrm{A}=\frac{136 \times 3}{34}=12 \\ & \mathrm{~B}=\frac{-5}{3}(12)=-20\end{aligned} I=∫0π/4A(3sin⁡x+5cos⁡x)3sin⁡x+5cos⁡x+∫0π/4B(3cos⁡x−5sin⁡x)3sin⁡x+5cos⁡x=A(x)0π/4+B[ln⁡(3sin⁡x+5cos⁡x)]0π/4=12(π4)−20ln⁡(32+52)−ln⁡(0+5)=3π−20ln⁡42+20ln⁡5=3π−20×52ln⁡2+20ln⁡5=3π−50ln⁡2+20ln⁡5\begin{aligned} & I=\int_0^{\pi / 4} \frac{A(3 \sin x+5 \cos x)}{3 \sin x+5 \cos x}+\int_0^{\pi / 4} \frac{B(3 \cos x-5 \sin x)}{3 \sin x+5 \cos x} \\ & =A(x)_0^{\pi / 4}+B[\ln (3 \sin x+5 \cos x)]_0^{\pi / 4} \\ & =12\left(\frac{\pi}{4}\right)-20 \ln \left(\frac{3}{\sqrt{2}}+\frac{5}{\sqrt{2}}\right)-\ln (0+5) \\ & =3 \pi-20 \ln 4 \sqrt{2}+20 \ln 5 \\ & =3 \pi-20 \times \frac{5}{2} \ln 2+20 \ln 5 \\ & =3 \pi-50 \ln 2+20 \ln 5\end{aligned}$
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