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Advanced techniques

Beta and gamma functions

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Γ(s)=∫0∞ts−1e−t dt,s>0\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}\,dt,\quad s>0

Recognize definite power-exponential integrals as gamma, or powers of t and 1−t on [0,1] as beta. For positive real parameters, B(p,q)=Γ(p)Γ(q)/Γ(p+q). Rescale before applying the formula.

Common mistake

Convergence depends on parameters; do not use the integral formula outside its hypotheses.

Step-by-step example

  1. Use t = 2x.

    ∫0∞x2e−2x dx=18∫0∞t2e−t dt\int_0^\infty x^2e^{-2x}\,dx=\frac18\int_0^\infty t^2e^{-t}\,dt

  2. Recognize Γ(3) and its recurrence.

    Γ(3)=2Γ(2)=2\Gamma(3)=2\Gamma(2)=2

  3. Adjust the scaling factor.

    I=Γ(3)/8=1/4I=\Gamma(3)/8=1/4

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