KCET · Maths · Statistics
If linear function \( f(x) \) and \( g(x) \) satisfy
\( \int[(3 x-1) \cos x+(1-2 x) \sin x] d x=f(x) \cos x+g(x) \sin x+C \), then
- A \( f(x)=3 x-5 \)
- B \( g(x)=3+x \)
- C \( f(x)=3(x-1) \)
- D \( g(x)=3(x-1) \)
Answer & Solution
Correct Answer
(D) \( g(x)=3(x-1) \)
Step-by-step Solution
Detailed explanation
Given that
\(I=\int[(3 x-1) \cos x+(1-2 x) \sin x] d x\)
\(=f(x) \cos x+g(x) \sin x+c\)
\(\Rightarrow I=(3 x-1) \sin x-\int \sin x \cdot 3 d x+(1-2 x)(-\cos x)+\int \cos x(-2) d x\)
\(=(3 x-1) \sin x+3 \cos x-\cos x+2 x \cos x-2 \sin x+c\)
\(=(3 x-1-2) \sin x+(2+2 x) \cos x+c\)
\(=3(x-1) \sin x+2(x+1) \cos x+c\)
Therefore, \(f(x)=2(x+1)\) and \(g(x)=3(x-1)\)
\(I=\int[(3 x-1) \cos x+(1-2 x) \sin x] d x\)
\(=f(x) \cos x+g(x) \sin x+c\)
\(\Rightarrow I=(3 x-1) \sin x-\int \sin x \cdot 3 d x+(1-2 x)(-\cos x)+\int \cos x(-2) d x\)
\(=(3 x-1) \sin x+3 \cos x-\cos x+2 x \cos x-2 \sin x+c\)
\(=(3 x-1-2) \sin x+(2+2 x) \cos x+c\)
\(=3(x-1) \sin x+2(x+1) \cos x+c\)
Therefore, \(f(x)=2(x+1)\) and \(g(x)=3(x-1)\)
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