Trigonometric substitution

In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions.^[1]^[2] Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.

Case I: Integrands containing a² − x²

Let $x=a\sin \theta ,$ and use the identity $1-\sin ^{2}\theta =\cos ^{2}\theta .$

Examples of Case I

Geometric construction for Case I

Example 1

In the integral

\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}},

we may use

x=a\sin \theta ,\quad dx=a\cos \theta \,d\theta ,\quad \theta =\arcsin {\frac {x}{a}}.

Then,

{\begin{aligned}\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}&=\int {\frac {a\cos \theta \,d\theta }{\sqrt {a^{2}-a^{2}\sin ^{2}\theta }}}\\[6pt]&=\int {\frac {a\cos \theta \,d\theta }{\sqrt {a^{2}(1-\sin ^{2}\theta )}}}\\[6pt]&=\int {\frac {a\cos \theta \,d\theta }{\sqrt {a^{2}\cos ^{2}\theta }}}\\[6pt]&=\int d\theta \\[6pt]&=\theta +C\\[6pt]&=\arcsin {\frac {x}{a}}+C.\end{aligned}}

The above step requires that $a>0$ and $\cos \theta >0.$ We can choose $a$ to be the principal root of $a^{2},$ and impose the restriction $-\pi /2<\theta <\pi /2$ by using the inverse sine function.

For a definite integral, one must figure out how the bounds of integration change. For example, as $x$ goes from $0$ to $a/2,$ then $\sin \theta$ goes from $0$ to $1/2,$ so $\theta$ goes from $0$ to $\pi /6.$ Then,

\int _{0}^{a/2}{\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\int _{0}^{\pi /6}d\theta ={\frac {\pi }{6}}.

Some care is needed when picking the bounds. Because integration above requires that $-\pi /2<\theta <\pi /2$ , $\theta$ can only go from $0$ to $\pi /6.$ Neglecting this restriction, one might have picked $\theta$ to go from $\pi$ to $5\pi /6,$ which would have resulted in the negative of the actual value.

Alternatively, fully evaluate the indefinite integrals before applying the boundary conditions. In that case, the antiderivative gives

\int _{0}^{a/2}{\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin \left({\frac {x}{a}}\right){\Biggl |}_{0}^{a/2}=\arcsin \left({\frac {1}{2}}\right)-\arcsin(0)={\frac {\pi }{6}}

as before.

Example 2

The integral

\int {\sqrt {a^{2}-x^{2}}}\,dx,

may be evaluated by letting ${\textstyle x=a\sin \theta ,\,dx=a\cos \theta \,d\theta ,\,\theta =\arcsin {\frac {x}{a}},}$ where $a>0$ so that ${\textstyle {\sqrt {a^{2}}}=a,}$ and ${\textstyle -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}}$ by the range of arcsine, so that $\cos \theta \geq 0$ and ${\textstyle {\sqrt {\cos ^{2}\theta }}=\cos \theta .}$

Then,

{\begin{aligned}\int {\sqrt {a^{2}-x^{2}}}\,dx&=\int {\sqrt {a^{2}-a^{2}\sin ^{2}\theta }}\,(a\cos \theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(1-\sin ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(\cos ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\[6pt]&=\int (a\cos \theta )(a\cos \theta )\,d\theta \\[6pt]&=a^{2}\int \cos ^{2}\theta \,d\theta \\[6pt]&=a^{2}\int \left({\frac {1+\cos 2\theta }{2}}\right)\,d\theta \\[6pt]&={\frac {a^{2}}{2}}\left(\theta +{\frac {1}{2}}\sin 2\theta \right)+C\\[6pt]&={\frac {a^{2}}{2}}(\theta +\sin \theta \cos \theta )+C\\[6pt]&={\frac {a^{2}}{2}}\left(\arcsin {\frac {x}{a}}+{\frac {x}{a}}{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\right)+C\\[6pt]&={\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+C.\end{aligned}}

For a definite integral, the bounds change once the substitution is performed and are determined using the equation ${\textstyle \theta =\arcsin {\frac {x}{a}},}$ with values in the range ${\textstyle -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}.}$ Alternatively, apply the boundary terms directly to the formula for the antiderivative.

For example, the definite integral

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

may be evaluated by substituting $x=2\sin \theta ,\,dx=2\cos \theta \,d\theta ,$ with the bounds determined using ${\textstyle \theta =\arcsin {\frac {x}{2}}.}$

Because $\arcsin(1/{2})=\pi /6$ and $\arcsin(-1/2)=-\pi /6,$

{\begin{aligned}\int _{-1}^{1}{\sqrt {4-x^{2}}}\,dx&=\int _{-\pi /6}^{\pi /6}{\sqrt {4-4\sin ^{2}\theta }}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}{\sqrt {4(1-\sin ^{2}\theta )}}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}{\sqrt {4(\cos ^{2}\theta )}}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}(2\cos \theta )(2\cos \theta )\,d\theta \\[6pt]&=4\int _{-\pi /6}^{\pi /6}\cos ^{2}\theta \,d\theta \\[6pt]&=4\int _{-\pi /6}^{\pi /6}\left({\frac {1+\cos 2\theta }{2}}\right)\,d\theta \\[6pt]&=2\left[\theta +{\frac {1}{2}}\sin 2\theta \right]_{-\pi /6}^{\pi /6}=[2\theta +\sin 2\theta ]{\Biggl |}_{-\pi /6}^{\pi /6}\\[6pt]&=\left({\frac {\pi }{3}}+\sin {\frac {\pi }{3}}\right)-\left(-{\frac {\pi }{3}}+\sin \left(-{\frac {\pi }{3}}\right)\right)={\frac {2\pi }{3}}+{\sqrt {3}}.\end{aligned}}

On the other hand, direct application of the boundary terms to the previously obtained formula for the antiderivative yields

{\begin{aligned}\int _{-1}^{1}{\sqrt {4-x^{2}}}\,dx&=\left[{\frac {2^{2}}{2}}\arcsin {\frac {x}{2}}+{\frac {x}{2}}{\sqrt {2^{2}-x^{2}}}\right]_{-1}^{1}\\[6pt]&=\left(2\arcsin {\frac {1}{2}}+{\frac {1}{2}}{\sqrt {4-1}}\right)-\left(2\arcsin \left(-{\frac {1}{2}}\right)+{\frac {-1}{2}}{\sqrt {4-1}}\right)\\[6pt]&=\left(2\cdot {\frac {\pi }{6}}+{\frac {\sqrt {3}}{2}}\right)-\left(2\cdot \left(-{\frac {\pi }{6}}\right)-{\frac {\sqrt {3}}{2}}\right)\\[6pt]&={\frac {2\pi }{3}}+{\sqrt {3}}\end{aligned}}

as before.

Case II: Integrands containing a² + x²

Let $x=a\tan \theta ,$ and use the identity $1+\tan ^{2}\theta =\sec ^{2}\theta .$

Examples of Case II

Geometric construction for Case II

Example 1

In the integral

\int {\frac {dx}{a^{2}+x^{2}}}

we may write

x=a\tan \theta ,\quad dx=a\sec ^{2}\theta \,d\theta ,\quad \theta =\arctan {\frac {x}{a}},

so that the integral becomes

{\begin{aligned}\int {\frac {dx}{a^{2}+x^{2}}}&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}+a^{2}\tan ^{2}\theta }}\\[6pt]&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}(1+\tan ^{2}\theta )}}\\[6pt]&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}\sec ^{2}\theta }}\\[6pt]&=\int {\frac {d\theta }{a}}\\[6pt]&={\frac {\theta }{a}}+C\\[6pt]&={\frac {1}{a}}\arctan {\frac {x}{a}}+C,\end{aligned}}

provided $a\neq 0.$

For a definite integral, the bounds change once the substitution is performed and are determined using the equation $\theta =\arctan {\frac {x}{a}},$ with values in the range $-{\frac {\pi }{2}}<\theta <{\frac {\pi }{2}}.$ Alternatively, apply the boundary terms directly to the formula for the antiderivative.

For example, the definite integral

\int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}\,

may be evaluated by substituting $x=\tan \theta ,\,dx=\sec ^{2}\theta \,d\theta ,$ with the bounds determined using $\theta =\arctan x.$

Since $\arctan 0=0$ and $\arctan 1=\pi /4,$

{\begin{aligned}\int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}&=4\int _{0}^{1}{\frac {dx}{1+x^{2}}}\\[6pt]&=4\int _{0}^{\pi /4}{\frac {\sec ^{2}\theta \,d\theta }{1+\tan ^{2}\theta }}\\[6pt]&=4\int _{0}^{\pi /4}{\frac {\sec ^{2}\theta \,d\theta }{\sec ^{2}\theta }}\\[6pt]&=4\int _{0}^{\pi /4}d\theta \\[6pt]&=(4\theta ){\Bigg |}_{0}^{\pi /4}=4\left({\frac {\pi }{4}}-0\right)=\pi .\end{aligned}}

Meanwhile, direct application of the boundary terms to the formula for the antiderivative yields

{\begin{aligned}\int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}\,&=4\int _{0}^{1}{\frac {dx}{1+x^{2}}}\\&=4\left[{\frac {1}{1}}\arctan {\frac {x}{1}}\right]_{0}^{1}\\&=4(\arctan x){\Bigg |}_{0}^{1}\\&=4(\arctan 1-\arctan 0)\\&=4\left({\frac {\pi }{4}}-0\right)=\pi ,\end{aligned}}

same as before.

Example 2

The integral

\int {\sqrt {a^{2}+x^{2}}}\,{dx}

may be evaluated by letting $x=a\tan \theta ,\,dx=a\sec ^{2}\theta \,d\theta ,\,\theta =\arctan {\frac {x}{a}},$

where $a>0$ so that ${\sqrt {a^{2}}}=a,$ and $-{\frac {\pi }{2}}<\theta <{\frac {\pi }{2}}$ by the range of arctangent, so that $\sec \theta >0$ and ${\sqrt {\sec ^{2}\theta }}=\sec \theta .$

Then,

{\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&=\int {\sqrt {a^{2}+a^{2}\tan ^{2}\theta }}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(1+\tan ^{2}\theta )}}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}\sec ^{2}\theta }}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int (a\sec \theta )(a\sec ^{2}\theta )\,d\theta \\[6pt]&=a^{2}\int \sec ^{3}\theta \,d\theta .\\[6pt]\end{aligned}}

The integral of secant cubed may be evaluated using integration by parts. As a result,

{\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)+C\\[6pt]&={\frac {a^{2}}{2}}\left({\sqrt {1+{\frac {x^{2}}{a^{2}}}}}\cdot {\frac {x}{a}}+\ln \left|{\sqrt {1+{\frac {x^{2}}{a^{2}}}}}+{\frac {x}{a}}\right|\right)+C\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {a^{2}+x^{2}}}+a^{2}\ln \left|{\frac {x+{\sqrt {a^{2}+x^{2}}}}{a}}\right|\right)+C.\end{aligned}}

Case III: Integrands containing x² − a²

Let $x=a\sec \theta ,$ and use the identity $\sec ^{2}\theta -1=\tan ^{2}\theta .$

Examples of Case III

Geometric construction for Case III

Integrals like

\int {\frac {dx}{x^{2}-a^{2}}}

can also be evaluated by partial fractions rather than trigonometric substitutions. However, the integral

\int {\sqrt {x^{2}-a^{2}}}\,dx

cannot. In this case, an appropriate substitution is:

x=a\sec \theta ,\,dx=a\sec \theta \tan \theta \,d\theta ,\,\theta =\operatorname {arcsec} {\frac {x}{a}},

where $a>0$ so that ${\sqrt {a^{2}}}=a,$ and $0\leq \theta <{\frac {\pi }{2}}$ by assuming $x>0,$ so that $\tan \theta \geq 0$ and ${\sqrt {\tan ^{2}\theta }}=\tan \theta .$

Then,

{\begin{aligned}\int {\sqrt {x^{2}-a^{2}}}\,dx&=\int {\sqrt {a^{2}\sec ^{2}\theta -a^{2}}}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int {\sqrt {a^{2}(\sec ^{2}\theta -1)}}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int {\sqrt {a^{2}\tan ^{2}\theta }}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int a^{2}\sec \theta \tan ^{2}\theta \,d\theta \\&=a^{2}\int (\sec \theta )(\sec ^{2}\theta -1)\,d\theta \\&=a^{2}\int (\sec ^{3}\theta -\sec \theta )\,d\theta .\end{aligned}}

One may evaluate the integral of the secant function by multiplying the numerator and denominator by $(\sec \theta +\tan \theta )$ and the integral of secant cubed by parts.^[3] As a result,

{\begin{aligned}\int {\sqrt {x^{2}-a^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)-a^{2}\ln |\sec \theta +\tan \theta |+C\\[6pt]&={\frac {a^{2}}{2}}(\sec \theta \tan \theta -\ln |\sec \theta +\tan \theta |)+C\\[6pt]&={\frac {a^{2}}{2}}\left({\frac {x}{a}}\cdot {\sqrt {{\frac {x^{2}}{a^{2}}}-1}}-\ln \left|{\frac {x}{a}}+{\sqrt {{\frac {x^{2}}{a^{2}}}-1}}\right|\right)+C\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}-a^{2}\ln \left|{\frac {x+{\sqrt {x^{2}-a^{2}}}}{a}}\right|\right)+C.\end{aligned}}

When ${\frac {\pi }{2}}<\theta \leq \pi ,$ which happens when $x<0$ given the range of arcsecant, $\tan \theta \leq 0,$ meaning ${\sqrt {\tan ^{2}\theta }}=-\tan \theta$ instead in that case.

Substitutions that eliminate trigonometric functions

Substitution can be used to remove trigonometric functions.

For instance,

{\begin{aligned}\int f(\sin(x),\cos(x))\,dx&=\int {\frac {1}{\pm {\sqrt {1-u^{2}}}}}f\left(u,\pm {\sqrt {1-u^{2}}}\right)\,du&&u=\sin(x)\\[6pt]\int f(\sin(x),\cos(x))\,dx&=\int {\frac {1}{\mp {\sqrt {1-u^{2}}}}}f\left(\pm {\sqrt {1-u^{2}}},u\right)\,du&&u=\cos(x)\\[6pt]\int f(\sin(x),\cos(x))\,dx&=\int {\frac {2}{1+u^{2}}}f\left({\frac {2u}{1+u^{2}}},{\frac {1-u^{2}}{1+u^{2}}}\right)\,du&&u=\tan \left({\tfrac {x}{2}}\right)\\[6pt]\end{aligned}}

The last substitution is known as the Weierstrass substitution, which makes use of tangent half-angle formulas.

For example,

{\begin{aligned}\int {\frac {4\cos x}{(1+\cos x)^{3}}}\,dx&=\int {\frac {2}{1+u^{2}}}{\frac {4\left({\frac {1-u^{2}}{1+u^{2}}}\right)}{\left(1+{\frac {1-u^{2}}{1+u^{2}}}\right)^{3}}}\,du=\int (1-u^{2})(1+u^{2})\,du\\&=\int (1-u^{4})\,du=u-{\frac {u^{5}}{5}}+C=\tan {\frac {x}{2}}-{\frac {1}{5}}\tan ^{5}{\frac {x}{2}}+C.\end{aligned}}

Hyperbolic substitution

Substitutions of hyperbolic functions can also be used to simplify integrals.^[4]

In the integral $\int {\frac {dx}{\sqrt {a^{2}+x^{2}}}}\,,$ make the substitution $x=a\sinh {u},$ $dx=a\cosh u\,du.$

Then, using the identities $\cosh ^{2}(x)-\sinh ^{2}(x)=1$ and $\sinh ^{-1}{x}=\ln(x+{\sqrt {x^{2}+1}}),$

{\begin{aligned}\int {\frac {dx}{\sqrt {a^{2}+x^{2}}}}\,&=\int {\frac {a\cosh u\,du}{\sqrt {a^{2}+a^{2}\sinh ^{2}u}}}\ ,\\[6pt]&=\int {\frac {a\cosh {u}\,du}{a{\sqrt {1+\sinh ^{2}{u}}}}}\,\\[6pt]&=\int {\frac {a\cosh {u}}{a\cosh u}}\,du\\[6pt]&=u+C\\[6pt]&=\sinh ^{-1}{\frac {x}{a}}+C\\[6pt]&=\ln \left({\sqrt {{\frac {x^{2}}{a^{2}}}+1}}+{\frac {x}{a}}\right)+C\\[6pt]&=\ln \left({\frac {{\sqrt {x^{2}+a^{2}}}+x}{a}}\right)+C\end{aligned}}

References

↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.
↑ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
↑ Stewart, James (2012). "Section 7.2: Trigonometric Integrals". Calculus - Early Transcendentals. United States: Cengage Learning. pp. 475–6. ISBN 978-0-538-49790-9.
↑ Boyadzhiev, Khristo N. "Hyperbolic Substitutions for Integrals" (PDF). Archived from the original (PDF) on 26 February 2020. Retrieved 4 March 2013.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.

[2] Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.

[:1-3] Stewart, James (2012). "Section 7.2: Trigonometric Integrals". Calculus - Early Transcendentals. United States: Cengage Learning. pp. 475–6. ISBN 978-0-538-49790-9.

[4] Boyadzhiev, Khristo N. "Hyperbolic Substitutions for Integrals" (PDF). Archived from the original (PDF) on 26 February 2020. Retrieved 4 March 2013.

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Integrals
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Case I: Integrands containing a2 − x2

Examples of Case I

Example 1

Example 2

Case II: Integrands containing a2 + x2

Examples of Case II

Example 1

Example 2

Case III: Integrands containing x2 − a2

Examples of Case III

Substitutions that eliminate trigonometric functions

Hyperbolic substitution

See also

References

Case I: Integrands containing a² − x²

Case II: Integrands containing a² + x²

Case III: Integrands containing x² − a²