U-Substitution — Mathwizurd

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U-Substitution

Basic idea

U-substitution is the reverse of the derivative chain rule. This is important when integrating an expression while chain rule is important while differentiating.

MathJax TeX Test Page Let's say you have the expression ddxesin(x)=cos(x)esin(x). We know this because of the chain rule. Now, let's go backward. esin(x)=cos(x)esin(x)dx In order to solve this, we have to use the opposite of the chain rule: u-substitution.

Notice that cos(x) is the derivative of sin(x). We can replace sin(x) with u(x). u(x)eu(x)dx We rearrange to write this: eu(x)u(x)dx Because it's the opposite of the chain rule, this equals eu(x)+C=esin(x)+C

Examples

MathJax TeX Test Page 2xcos(x2)dx In order to solve this, we have to look for some chain rule. So, we look inside our wrapping function, cos, which envelops x2. Now, we say u=x2du=2xdx We rewrite our integral as cos(u)dusin(u)+C=sin(x2)+C
MathJax TeX Test Page sin5(x)cos5(x)dx sin5(x)(1sin2(x))2cos(x)dx We now set u=sin(x), so du=cos(x)dx. u5(1u2)2du u5(u42u2+1)du=u92u7+u5du=u1010u84+u66+C Our final answer becomes sin10(x)10sin8(x)4+sin6(x)6+C
MathJax TeX Test Page cos(10x+5)dx This problem requires u-substitution, even though there is only one term. This is because the integral isn't sin(10x+5)! If we differentiate that we get 10cos(10x+5) So, how do we u-sub this integral? We create the u(x)! We multiple by 1010. cos(10x+5)dx=11010cos(10x+5)dx Now, we set u(x)=10x+5,du=10dx. The integral becomes 110cos(u)du110sin(10x+5)+C
David Witten

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