TS EAMCET · Maths · Indefinite Integration
If \(I_m=\int x^m \cos n x d x=g(x)-\frac{m(m-1)}{n^2} I_{m-2}\), then \(g(x)=\)
- A \(\frac{x^m \sin n x}{n}+\frac{m(m-1) x^{m-1} \cos n x}{n^2}\)
- B \(\frac{x^m \cos n x}{n}+\frac{x^{m-1} m(m-1)}{n^2} \sin n x\)
- C \(\frac{m}{n} \sin n x+\frac{m}{n^2} x^{m-1} \cos n x\)
- D \(\frac{x^m \sin n x}{n}+\frac{m}{n^2} x^{m-1} \cos n x\)
Answer & Solution
Correct Answer
(D) \(\frac{x^m \sin n x}{n}+\frac{m}{n^2} x^{m-1} \cos n x\)
Step-by-step Solution
Detailed explanation
Given that, \(I_m=\int x^m \cos n x d x=g(x)-\frac{m(m-1)}{n^2} I_{m-2}\) \(\ldots(i)\) \(I_{m-2}=\int x^{m-2} \cos n x d x\) Solving integration by parts \(\left.=\cos n x \int x^{m-2} d x-\int\left(\frac{d}{d x} \cos n x\right) \int x^{m-2} d x\right) d x\)…
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