TS EAMCET · Maths · Definite Integration
\(\int_0^{\pi / 4} \frac{\sec x}{3 \cos x+4 \sin x} d x=\)
- A \(\log \left(\frac{7}{3}\right)\)
- B \(\frac{1}{4} \log \left(\frac{7}{3}\right)\)
- C \(\frac{1}{4} \log 7\)
- D log 7
Answer & Solution
Correct Answer
(B) \(\frac{1}{4} \log \left(\frac{7}{3}\right)\)
Step-by-step Solution
Detailed explanation
\(I = \int_0^{\pi / 4} \frac{\sec^2 x}{3 + 4 \tan x} d x\) Let \(u = 3 + 4 \tan x\). Then \(d u = 4 \sec^2 x d x\). Limits: \(x=0 \implies u=3\), \(x=\pi/4 \implies u=7\). \(I = \frac{1}{4} \int_3^7 \frac{1}{u} d u\) \(I = \frac{1}{4} [\ln |u|]_3^7\)…
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