Integral Calculus Formula Sheet

1 Integral Calculus Formula Sheet Derivative Rules: 0dcdx 1nndxnxdx sincosdxxdx secsec tandxxxdx 2tansecdxxdx cossindxxdx csccsc cotdxxxdx 2cotcscdxxdx lnxxdaaadx xxdeedx ddcf xcf xdxdx dddf x gxf xgxdxdxdx fg f g fg 2fg fgfgg dfgxf gx gxdx Properties of Integrals: ()()kf u du k f u du () ()()()fu gu duf udugudu ()0aafxdx ()()baabfxdxf xdx ()()()cbcaabfxdxf xdxf xdx 1()baveaffxdxba 0()2 ()aaafxdxf xdx if f(x) is even ()0aafxdx if f(x) is odd ()()( ()) ()()fbbafagfx f xdxgudu udv uvvdu Integration Rules: du u C 11nnuuduCn lnduuCu uuedu e C 1lnuuadua Ca sincosuduu C cossinuduu C 2sectanuduu C 2csccotuuC csc cotcscuuduuC sec tansecuudu uC 221arctanduuCau aa 22arcsinduuCaau 221secuduarcCaauu a Fundamental Theorem of Calculus : ' xadFxftdtfxdx where ft is a continuous function on [a, x]. bafxdx Fb F a, where F(x) is any antiderivative of f(x).

2 Riemann Sums: 11nniiiicaca 111nnniiiiiiiabab 1()lim()bnniafxdxf a i x x nabx 11nin 1(1)2ninni 21(1)(21)6ninnni 231(1)2ninni height of th rectanglewidth of th rectangleiii Right Endpoint Rule: ninabnabniiafxxiaf1)()(1)()()()( Left Endpoint Rule: ()()11((1))() ( )((1) )nnbabanniifa ix xfa i Midpoint Rule: (1)( )(1)( )2211()()()( )nniibaiibanniifax xfa Net Change: displacement : ()bavxdx Distance Traveled: ()bavx dx 0()(0)( )tst svxdx 0()(0)( )tQt QQ xdx Trig Formulas: 212sin ( )1 cos(2 )xx sintancosxxx 1seccosxx cos()cos( )xx 22sin ( ) cos ( ) 1xx 212cos ( )1 cos(2 )xx coscotsinxxx 1cscsinxx sin()sin( )xx 22tan ( ) 1 sec ( )xx Geometry Fomulas: Area of a Square: 2As Area of a Triangle: 12 Abh Area of an Equilateral Trangle:234As Area of a Circle: 2Ar Area of a Rectangle: Abh Areas and Volumes: Area in terms of x (vertical rectangles): ()batop bottom dx Area in terms of y (horizontal rectangles): ()dcright left dy General Volumes by Slicing: Given: Base and shape of Cross sections ()baVAxdx if slices are vertical ()dcVAydy if slices are horizontal Disk Method: For volumes of revolution laying on the axis with slices perpendicular to the axis 2()baVRxdx if slices are vertical 2()dcVRydy if slices are horizontal Washer Method: For volumes of revolution not laying on the axis with slices perpendicular to the axis 22()()baVRx rxdx if slices are vertical 22()()dcVRy rydy if slices are horizontal Shell Method: For volumes of revolution with slices parallel to the axis 2baVrhdx if slices are vertical 2dcVrhdy if slices are horizontal Physical Applications: Physics Formulas Associated Calculus Problems Mass.

3 Mass = Density * Volume (for 3 D objects) Mass = Density * Area (for 2 D objects) Mass = Density * Length (for 1 D objects) Mass of a one dimensional object with variable linear density: ()()bbdistanceaaMasslinear density dxx dx Work: Work = Force * Distance Work = Mass * Gravity * Distance Work = Volume * Density * Gravity * Distance Work to stretch or compress a spring (force varies): '()()bbbHooke s Lawaaafor springsWorkforce dxF x dxkxdx Work to lift liquid: ()()( )() * * ( ) * ()()dcvolumedcWorkgravity density distance area of a slice dyWAyaydyinmetric Force/Pressure: Force = Pressure * Area Pressure = Density * Gravity * Depth Force of water pressure on a vertical surface: ()()()() * * () * ( )()dcareadcForcegravity density depth width dyFaywydyinmetric Integration by Parts: Knowing which function to call u and which to call dv takes some practice. Here is a general guide: u Inverse Trig Function (1sin, arccos ,xx etc) Logarithmic Functions (log 3 , ln(1),xx etc) Algebraic Functions (3,5,1/,xxx etc) Trig Functions (sin(5 ), tan( ),xxetc) dv Exponential Functions (33,5 ,xxeetc) Functions that appear at the top of the list are more like to be u, functions at the bottom of the list are more like to be dv.

4 Trig Integrals: Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x): 1. If the power of the sine is odd and positive: Goal: cosux i. Save a sin( )dux dx ii. Convert the remaining factors to cos( )x(using 22sin1 cosxx .) 1. If the power of sec( )xis even and positive: Goal:tanux i. Save a 2sec ( )dux dx ii. Convert the remaining factors to tan( )x (using 22sec1 tanxx .) 2. If the power of the cosine is odd and positive:Goal:sinux i. Save a cos( )dux dx ii. Convert the remaining factors to sin( )x(using 22cos1 sinxx .) 2. If the power of tan( )xis odd and positive: Goal:sec( )ux i. Save a sec( ) tan( )duxx dx ii. Convert the remaining factors to sec( )x (using 22sec1tanxx .) 3. If both sin( )x and cos( )xhave even powers: Use the half angle identities: i. 212sin ( )1 cos(2 )xx ii. 212cos ( )1 cos(2 )xx If there are no sec(x) factors and the power of tan(x) is even and positive, use 22sec1tanxx to convert one 2tanxto 2secx Rules for sec(x) and tan(x) also work for csc(x) and cot(x) with appropriate negative signsIf nothing else works, convert everything to sines and cosines.

5 Trig Substitution: Expression Substitution Domain Simplification 22au sinua 22 22cosau a 22au tanua 22 22secau a 22ua secua 0,2 22tanua a Partial Fractions: Linear factors: Irreducible quadratic factors:211111 1()..()()() () ()mmmPxABYZ xrxrxrxr xr 22222121111 1()..()()() ()()mmmPxAxBCxD WxXYxZxrxrxrxr xr If the fraction has multiple factors in the denominator, we just addthe decompositions.