MHT CET · Maths · Indefinite Integration
If \(\int \frac{3 \sin x \cos x}{4 \sin x+7} \mathrm{~d} x=\mathrm{A} \sin x-\mathrm{B} \log (4 \sin x+7)+\mathrm{c}\) where \(c\) is the constant of integration, then the value of \(A+B\) is equal to
- A \(\frac{9}{16}\)
- B \(\frac{-9}{16}\)
- C \(\frac{33}{16}\)
- D \(\frac{-33}{16}\)
Answer & Solution
Correct Answer
(C) \(\frac{33}{16}\)
Step-by-step Solution
Detailed explanation
Let \(t = 4 \sin x + 7\). Then \(\mathrm{d} t = 4 \cos x \, \mathrm{d} x \Rightarrow \cos x \, \mathrm{d} x = \frac{1}{4} \mathrm{d} t\) and \(\sin x = \frac{t-7}{4}\). \(\int \frac{3 \left(\frac{t-7}{4}\right) \frac{1}{4} \mathrm{d} t}{t} = \frac{3}{16} \int \left(1 - \frac{7}{t}\right) \mathrm{d} t\)
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