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int_(0)^(1)cos ((pi)/(1-x))*(dx)/((1-x)^(2)).

∫01cos⁡π1−x⋅dx(1−x)2 \int_{0}^{1} \cos \frac{\pi}{1-x} \cdot \frac{d x}{(1-x)^{2}} .

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Evaluate definite integrals using the chain ruleright chevron icon

Full solution

Q. ∫01cos⁡π1−x⋅dx(1−x)2 \int_{0}^{1} \cos \frac{\pi}{1-x} \cdot \frac{d x}{(1-x)^{2}} .
  1. Substitution step: Let's do a substitution: let u=1−xu = 1 - x, then du=−dxdu = -dx.
  2. Change limits: Change the limits of integration. When x=0x = 0, u=1u = 1. When x=1x = 1, u=0u = 0.
  3. Substitute in terms of uu: Substitute everything in terms of uu. The integral becomes −∫10cos⁡(π/u)u−2 du-\int_{1}^{0}\cos(\pi/u)u^{-2}\,du.
  4. Flip limits: Flip the limits of integration to get rid of the negative sign. The integral is now ∫01cos⁡(π/u)u−2 du\int_{0}^{1}\cos(\pi/u)u^{-2}\,du.
  5. Integrate with respect to uu: Now, integrate cos⁡(πu)u−2\cos(\frac{\pi}{u})u^{-2} with respect to uu. This is not a standard integral and cannot be solved using elementary functions.

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