Tangent half-angle substitution

Builds on Integration by substitution Sine and cosine

Summary

The tangent half-angle substitution sets $t = \tan(x/2)$ in an integral. It turns each rational expression in $\sin x$ and $\cos x$ into a rational function of $t$ . Partial fractions then finish the computation.

Remarks

The substitution gives $\sin x = \frac{2t}{1 + t^2}$ , $\cos x = \frac{1 - t^2}{1 + t^2}$ and $dx = \frac{2 \, dt}{1 + t^2}$ . It works on intervals where $x/2$ stays inside $(-\pi/2, \pi/2)$ . For example, it gives $\int \frac{dx}{\sin x} = \ln |\tan(x/2)| + C$ .

Reading path

To understand this page from the ground up, read these 56 pages in order. The foundations come first.

Limit point of a set definition
Local maximum definition
Partition of an interval definition
Sequence definition
Limit of a function definition
Local minimum definition
Mesh of a partition definition
Riemann sum definition
Real sequence definition
Subsequence definition
Continuity at a point definition
Derivative definition
Riemann integrable function definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Continuous function definition
Epsilon-delta definition of continuity statement
Antiderivative definition
Differentiable function definition
Fermat's theorem on stationary points statement
Riemann integral definition
Upper and lower Darboux sums definition
Convergent sequence definition
Monotone convergence theorem for sequences statement
Every sequence has a monotone subsequence statement
Power series definition
Intermediate value theorem statement
Composition of continuous functions is continuous statement
Uniform continuity definition
Derivative function definition
Definite integral definition
Upper and lower Darboux integrals definition
Bolzano–Weierstrass theorem statement
Sine and cosine definition
Chain rule statement
Uniform continuity on a set definition
Boundedness theorem statement
Higher derivative definition
Monotonicity of the integral statement
Additivity of the integral over intervals statement
Darboux and Riemann integrals agree statement
Extreme value theorem statement
Continuously differentiable function definition
Continuous functions on a closed bounded interval are uniformly continuous statement
Riemann's criterion for integrability statement
Mean value theorem for integrals statement
Rolle's theorem statement
Continuous functions are Riemann integrable statement
Mean value theorem statement
First fundamental theorem of calculus statement
Second fundamental theorem of calculus statement
The integral of a continuous function is an antiderivative statement
Integration by substitution statement
Tangent half-angle substitution concept

Tag	`0W8ETY2AZ6`
URI	`m2://0W8ETY2AZ6`
Aliases	`weierstrass-substitution`
Kind	concept
Writing	W1 stated
MSC	26A36
Uses	Integration by substitution, Sine and cosine
Same as	Wikidata Q663628 Wikipedia Tangent half-angle substitution

Tangent half-angle substitution

Summary

Remarks

Reading path

Further reading

What this page needs for W2

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