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Integrate the function $\sin x\sin(\cos x)$

$\begin{array}{1 1} \cos(\cos x)+c. \\\cos(\sin x)+c. \\ \sin(\cos x)+c. \\ \sin(\sin x)+c.\end{array} $

1 Answer

  • Method of substitution:
  • Given $\int f(x)dx$ can be transformed into another form by changing independent variable x to t by substituting x=g(t).
  • Consider $I=\int f(x)dx.$
  • Put x=g(t) so that $\frac{dx}{dx}=g'(t).
  • dx=g'(t)dt.
  • Thus $ I=\int f(g(t).g'(t))dt.$
Given $I=\int\sin x\sin(\cos x)dx.$
Let $\cos x$ be t.
Differentiating on both sides we get,
$-\sin xdx=dt$.
Now substituting for $\cos x$ and $\sin xdx$ we get,
$I=-\int \sin t.dt$
On integrating we get,
$\cos t+c$.
Now Substituting for t we get,
$\cos(\cos x)+c.$
Hence $\int\sin x\sin(\cos x)dx=\cos(\cos x)+c.$


answered Jan 28, 2013 by sreemathi.v