By Zac Cockman Liz Mooney Integrals By Zac Cockman Liz Mooney

Integration Techniques Integration is the process of finding an indefinite or diefinite integral Integral is the definite integral is the fundamental concept of the integral calculus. It is written as Where f(x) is the integrand, a and b are the lower and upper limits of integration, and x is the variable of integration.

Integration techniques Integration is the opposite of Differentiation. Power Rule U-Substitution Special Cases Sin and Cos

Power Rule n cannot equal -1 u=x Du=dx N=1 C = constant + c

Examples U = 2x Du = dx n = 1 Answer = X2 + C

Examples Answer = X3/3 + 3x2/2 + 2x + C U=x Du=dx N=2 N=1 N=0

Examples U = 4 X2 Du = 8xdx N = -1/2 3/8 * 2 * (4x2 + 5)1/2 + C Answer ¾(4X2 + 5)1/2 + C

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Try Me Continue U = 1 +x2 Du = 2dx N = -1/2 1/2 [2(1+x2)1/2] + C

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Try Me U = x4 + 3 Du = 4x3 dx N = 2

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U-Sub What is U-Sub When do you use it Steps Find your u, du, and for u, solve for x Replace all the x for u. Do the same steps for power rule At the end replace the u in the problem for your u when you found it in the beginning.

Example U= X= u2 -1 dx= 2udu (u2 – 1) u(2udu) 2u4 – 2u2 2/5 (u5 – 2/3u3) + c 2/5 (x+1) 5/2 + c

Example U = U2 – 1 = x 2udu = dx 2/3(x+1)3/2 -2(x+1) + c

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Try Me U = X = Du = udu 1/10 u5 + 1/2u2 + c 1/10 (2x-3)5/2 + ½(2x-3) + c

Special Cases When n = -1 the u is put inside the absolute value of the natural log If there is only one x in the problem and it is squared, square the term before taking the interval

Special Cases Examples U = x-1 Du = dx N = -1

Special Cases Examples

Integration using Powers of Sin and Cos Three Methods Odd-Even Odd-Odd Even-Even In Odd-Even, take the odd power and re write the odd power as odd even Re write the even power change it using Pythagorean identity. In Odd-Odd, take one of the odds, change to odd even Use same rules

Integration using Powers of sin and cos For Even-Even, change the power to the half angle formula. Special Case If the Power of the trig is 1, u is the angle

Powers of Trig Odd - Even Take the odd power, re write the odd power as odd even Re write the even power, change it using the Pythagorean identities. ∫sin5xcos4xdx ∫sin4x sinxcos4xdx ∫(1-cos2x)2 sinxcos2xdx

Powers of Trig Odd-Even ∫(1-2cos2x+cos4x) sinxcos4xdx ∫sinxcos4xdx-2 ∫sinx cos6xdx+∫sinxcos8xdx U = cosx Du = -sinx N = 4 N = 6 N = 8 -1/5cos5x+2/7cos7x-1/9cos9x+c

Try Me ∫sin32xcos22xdx

Powers of Trig Odd - Even Try Me ∫sin32xcos22xdx ∫sin22xsin2xcos22xdx ∫(1-cos22x) sin2xcos22xdx -1/2 ∫sin22xcos22xdx+1/2 ∫sin2xcos42xdx U = cosx Du = -2sin2x N = 2 N = 4 -1/6 cos32x+1/10cos52x+c

Powers of Trig Odd Odd Take one of the odds, change to odd even. Use other rules to finish. Example

Powers of trig Odd-Odd U = cosx du= -sinxdx n = 3 n = 5

Powers of Trig Odd-Odd Example

Powers of Trig Odd Odd Example Continued U = cosx Du = -sinx N = 17

Powers of Trig Even-Even Change to half angle formula ∫sin2xdx ∫1-cos2xdx 2 1/2∫dx-(1/2)(1/2)∫2cos2xdx U = x U = 2x Du = dx Du = 2dx 1/2x-1/4sin2x+c

Try Me ∫sin2xcos2x

Powers of Trig Even-Even Try Me ∫sin2xcos2x ∫(1-cos2x)(1+cos2x)        2            2 1/4∫(1-cos22x)dx 1/4∫sin22xdx 1/4∫1-cos4x/2dx 1/8∫dx-(1/4)(1/8) ∫4cos4xdx 1/8x-1/32sin4x+c

Solving for Integrals U =x-1 Du = dx N = 2 9 – 0 = 9

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Solving for Integrals U = x2 + 2 Du = 2xdx N = 2

Bibliography www.musopen.com Mathematics Dictionary, Fourth Edition, James/James, Van Nostrand Reinnhold Company Inc., 1976