Skip to content
MathSolver Pro
  • Open the math solver
  • Ask the AI tutor
  • Find an AP course
  • Create a quiz
  • Open the calculator
  • Graphing Calculator
  • Browse all calculators
  • Read math guides
↑ ↓Enter to open · Esc to close
Solve a problem
Indefinite integralsCalculusCollegeabout 0:53

Integral of x cos(x): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x·sin(x) + cos(x) + C.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫xcos⁡(x) dx\int x \cos\left( x\right)\,dx
  2. 2. Step by step
    1. ∫xcos⁡(x) dx\int x \cos\left( x\right)\,dx
      Start with the integral
    2. ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
      Integration by parts (tabular method)
    3. ∫xcos⁡(x) dx=xsin⁡(x)+cos⁡(x)\int x \cos\left( x\right)\,dx = x \sin\left( x\right)+\cos\left( x\right)
      Combine the alternating products
    4. ∫xcos⁡(x) dx=xsin⁡(x)+cos⁡(x)+C\int x \cos\left( x\right)\,dx = x \sin\left( x\right)+\cos\left( x\right) + C
      Simplify and add the constant of integration
  3. 3. The answer
    xsin⁡(x)+cos⁡(x)+C x \sin\left( x\right)+\cos\left( x\right) + C

    ∫ x cos(x) dx = x·sin(x) + cos(x) + C

  4. 4. Check
    • Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integral of x cos(x). Find the indefinite integral (antiderivative) of x cos(x) with respect to x.

Step by step. Start with the integral. Integration by parts (tabular method). Differentiate the polynomial x until it reaches 0, integrate cos(x) the same number of times, and combine with alternating signs. Combine the alternating products. Simplify and add the constant of integration.

The answer. So the answer is x times sin(x) plus cos(x) plus C. An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

Check. Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.

About indefinite integrals

Find antiderivatives with standard rules, substitution and integration by parts.

More lessons like this

Integral of x cos(x): Animated Step-by-Step Lesson (College)