Integrate sin(4x)x: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int \sin\left(4x\right) x\,dx$
2. 2. Step by step
   1. $$\int \sin\left(4x\right) x\,dx$$
      Start with the integral
   2. $$\int u\,dv = uv - \int v\,du$$
      Integration by parts (tabular method)
   3. $$\int \sin\left(4x\right) x\,dx = -\frac{x \cos\left(4x\right)}{4}+\frac{\sin\left(4x\right)}{16}$$
      Combine the alternating products
   4. $$\int \sin\left(4x\right) x\,dx = -\frac{x \cos\left(4x\right)}{4}+\frac{\sin\left(4x\right)}{16} + C$$
      Simplify and add the constant of integration
3. 3. The answer
   $$-\frac{x \cos\left(4x\right)}{4}+\frac{\sin\left(4x\right)}{16} + C$$

   ∫ sin(4x) x dx = -(x·cos(4x) / 4) + sin(4x) / 16 + C

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

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

• Video lesson

  $\int \frac{1}{\sqrt{4-{x}^{2}}}\,dx$

  ▶0:44
  CalculusCollege3 steps

  Evaluate ∫ 1/√(4 - x² dx

  Indefinite integrals

• Video lesson

  $\int \ln\left( x\right)\,dx$

  ▶0:38
  CalculusCollege3 steps

  Antiderivative of ln(x)

  Indefinite integrals

• Video lesson

  $\int {x}^{\frac{2}{3}}\,dx$

  ▶0:44
  CalculusCollege3 steps

  Antiderivative of x^(2/3)

  Indefinite integrals

• Video lesson

  $\int \sin\left(3x\right) x\,dx$

  ▶0:50
  CalculusCollege4 steps

  Integrate sin(3x)x

  Indefinite integrals

• Video lesson

  $\int \frac{1}{\left( x+1\right) \left( x+4\right)}\,dx$

  ▶0:45
  CalculusCollege3 steps

  Integrate 1/((x+1)(x+4))

  Indefinite integrals

• Video lesson

  $\int \frac{1}{{x}^{2}+1}\,dx$

  ▶0:42
  CalculusCollege3 steps

  Integrate 1/(x²+1)

  Indefinite integrals

• All indefinite integrals lessons
• Browse the whole lesson library
• Make a lesson from your own problem
• Step-by-step solver: check your own answer.