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→Methods of Integration
===Algebraic Substitution===
===Algebraic Substitution===
'''Integration by Algebraic Substitution''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by substituting a more complicated expression with a variable.
Given an integral
:<big><math>\int\frac{2x}{x^2+3}dx</math></big>
We can substitute the term :<big><math>x^2+3</math></big> with a u. Giving us
:<big><math>u=x^2+3</math></big>
We then take the derivative of u with respect to x,
:<big><math>\frac{du}{dx}x^2+3=2x</math></big>
We then set the terms equal to du,
:<big><math>du=2xdx</math></big>
Now we are ready to rewrite the integral,
:<big><math>\int\frac{2x}{x^2+3}dx=\int\frac{1}{u}du</math></big>
We can rewrite the intergal this way due to the subsitution of the x terms with the u terms.
Now we can solve the intergral in terms of u.
:<big><math>\int\frac{1}{u}du=ln|u|+c</math></big>
Now we replace u with the term :<big><math>x^2+3</math></big> to get,
:<big><math>ln|x^2+3|+c</math></big>
We can check this by taking the derivative of :<big><math>ln|x^2+3|</math></big>,
:<big><math>\frac{d}{dx}ln|x^2+3|=(\frac{1}{x^2+3})(2x)=\frac{2x}{x^2+3}</math></big>
===Trigonometric Substitution===
===Trigonometric Substitution===