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int x cos xdx

∫xcos⁡xdx \int x \cos x d x

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Find indefinite integrals using the substitution and by partsright chevron icon

Full solution

Q. ∫xcos⁡xdx \int x \cos x d x
  1. Choose uu and dvdv: We will use integration by parts to solve the integral of xcos⁡(x) dxx \cos(x) \, dx. Integration by parts is given by the formula ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du, where uu and dvdv are parts of the integrand that we choose. We will let u=xu = x and dv=cos⁡(x) dxdv = \cos(x) \, dx.
  2. Find dudu and vv: Differentiate uu to find dudu, and integrate dvdv to find vv. So, du=dxdu = dx and v=sin⁡(x)v = \sin(x). We can now apply the integration by parts formula.
  3. Apply integration by parts: Substitute uu, vv, and dudu into the integration by parts formula to get ∫xcos⁡(x) dx=xsin⁡(x)−∫sin⁡(x) dx\int x \cos(x) \, dx = x \sin(x) - \int\sin(x) \, dx.
  4. Integrate sin⁡(x)\sin(x): Now we need to integrate ∫sin⁡(x) dx\int \sin(x) \, dx, which is a basic integral. The integral of sin⁡(x)\sin(x) with respect to xx is −cos⁡(x)-\cos(x).
  5. Substitute sin⁡(x)\sin(x) integral: Substitute the integral of sin⁡(x)\sin(x) into the equation from the previous step to get ∫xcos⁡(x) dx=xsin⁡(x)−(−cos⁡(x))\int x \cos(x) \, dx = x \sin(x) - (-\cos(x)).
  6. Simplify the equation: Simplify the equation by distributing the negative sign to get ∫xcos⁡(x) dx=xsin⁡(x)+cos⁡(x)+C\int x \cos(x) \, dx = x \sin(x) + \cos(x) + C, where CC is the constant of integration.

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