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Solve the following.
Evaluate: ∫ 1/(x²)sin²(1/x) dx
Evaluate:
$\int \frac{1}{x^2}\sin^2\left(\frac{1}{x}\right) dx$
Answer
Answer
Integral: $\int \left(sin(1/x)^2/x^2\right) dx$
Put $t = \frac{1}{x}$, so $dt = -\frac{1}{x^2}dx$.
$\int\frac{1}{x^2}\sin^2\left(\frac{1}{x}\right)dx = -\int\sin^2 t\,dt = -\int\frac{1-\cos 2t}{2}\,dt$
$= -\frac{t}{2} + \frac{\sin 2t}{4} + C$
Hence the integral is $-\frac{1}{2x} + \frac{1}{4}\sin\left(\frac{2}{x}\right) + C$.
Final answer: $-\frac{1}{2x} + \frac{1}{4}\sin\left(\frac{2}{x}\right) + C$