Lecture Week 4 - Calculus 2 MATH 1230 Kwantlen University Inverse Trigs 7.1 Integration By Parts October 2 2018 1 11 Integrals as Inverse

# Lecture Week 4 - Calculus 2 MATH 1230 Kwantlen...

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Calculus 2: MATH 1230Kwantlen UniversityInverse Trigs7.1 Integration By PartsOctober 2, 20181 / 11
Integrals as Inverse Trigonometric FunctionsFunctionRecall derivative?Therefore the integral,sin-1(x)11-x211-x2dx=sin-1(x)cos-1(x)-11-x2tan-1(x)11+x211+x2dx=tan-1(x)cot-1(x)-11+x2sec-1(x)1xx2-11xx2-1dx=sec-1(x)csc-1(x)-1xx2-1Now use these,u-sub and creativity,to take care of more general integrals.2 / 11
Example:116-x2dxSolution:Letu=x4orx=4u, therefore,dx=4du.116-x2dx=116-16u24du=11-u2du=arcsinu+C=arcsinx4+CExample:14+x2dxKey: 1 2 arctan(x2)+C3 / 11
More Examples:1.ex1+e2xdxKey: arctan(ex)+C2.dxtan-1x(1+x2)Hint: Useu-subKey: ln(arctanx)+C4 / 11
Example (Challenging)1.dx5+4x-x2Hint:Complete the square 5+4x-x2=9-(x-2)2.Key: arcsin(x-2)3+C5 / 11
Integration By Parts6 / 11
Method 2: Integration By PartsWhenever you see an integral involving amix ofex,lnx