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Integration techniques

The t = tan(x/2) substitution

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

sin⁡x=2t1+t2,cos⁡x=1−t21+t2\sin x=\frac{2t}{1+t^2},\quad\cos x=\frac{1-t^2}{1+t^2}

Transform a rational function of sin x and cos x into a rational function of t. Use dx=2dt/(1+t²) and rational integration techniques. Work on intervals where tan(x/2) is defined.

Common mistake

The substitution is general for rational trigonometric functions but may produce a long fraction.

Step-by-step example

  1. Introduce t and transform 1+cos x.

    ∫dx1+cos⁡x,1+cos⁡x=21+t2\int\frac{dx}{1+\cos x},\quad1+\cos x=\frac2{1+t^2}

  2. Substitute dx too.

    I=∫2/(1+t2)2/(1+t2) dt=∫dtI=\int\frac{2/(1+t^2)}{2/(1+t^2)}\,dt=\int dt

  3. Return to x and retain exclusions.

    I=tan⁡(x/2)+CI=\tan(x/2)+C

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