Fandom

Math Wiki

Integration by trigonometric substitution

Redirected from Trigonometric substitution

1,078pages on
this wiki
Add New Page
Talk0 Share

Integration by trigonometric substitution is a technique of integration that involves substituting some function of x for a trigonometric function.

As a general rule, when taking an antiderivative of a function in the form \sqrt{a^2-x^2} , the substitution x=a\sin(u) is usually the best option. For \sqrt{a^2+x^2} and \sqrt{a^2-x^2} , the substitutions x=a\tan(u) and x=a\sec(u) (respectively) are usually the best options.

Examples

This technique can be used when functions would be otherwise difficult to integrate. One of the most well-known examples is

\int\dfrac{dx}{\sqrt{1-x^2}}

Here, we can use the substitution x=\sin(u),\ dx=\cos(u)du to get

\begin{align}&\int\dfrac{\cos(u)}{\sqrt{1-\sin^2(u)}}du=\int\dfrac{\cos(u)}{\sqrt{\cos^2(u)}}du=\int du=u\\&x=\sin(u)\Rightarrow\arcsin(x)=u\end{align}

Therefore:

\int\dfrac{dx}{\sqrt{1-x^2}}=\arcsin(x)

A second example:

\int\dfrac{dx}{x^2\sqrt{4+x^2}}=\int\dfrac{dx}{x^2\sqrt{2^2+x^2}}

Here, we can use the substitution x=2\tan(u),\ dx=2\sec^2(u)du to get

\begin{align}\int\dfrac{2\sec^2(u)}{(2\tan(u))^2\sqrt{4(1+\tan^2(u))}}du&=\int\dfrac{2\sec^2(u)}{8\tan^2(u)\sqrt{1+\tan^2(u)}}du\\&=\frac14\int\dfrac{\sec^2(u)}{\tan^2(u)\sqrt{1+\tan^2(u)}}du\end{align}

By using the trigonometric identity 1+\tan^2(u)=\sec^2(u) , we get

\begin{align}\frac14\int\dfrac{\sec^2(u)}{\tan^2(u)\sqrt{\sec^2(u)}}du&=\frac14\int\dfrac{\sec^2(u)}{\tan^2(u)\sec(u)}du\\&=\frac14\int\frac{\sec(u)}{\tan^2(u)}du=\frac14\int\frac{\cos(u)}{\sin^2(u)}du\end{align}

Which evaluates to -\frac{1}{\sin(u)} by using u-substitution. Since

x=2\tan(u)\Rightarrow\arctan\left(\tfrac{x}{2}\right)=u

we can say that

-\frac{1}{4\sin(u)}=-\frac{1}{4\sin\left(\arctan\left(\tfrac{x}{2}\right)\right)}
=-\frac{1}{\dfrac{4x}{\sqrt{4+x^2}}}=-\frac{\sqrt{4+x^2}}{4x}

Therefore:

\int\dfrac{dx}{x^2\sqrt{4+x^2}}=-\frac{\sqrt{4+x^2}}{4x}+C

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Also on Fandom

Random Wiki