15.1 Notes Antiderivatives and Indefinite Integrals

Name Period Honors Brief Calc. 15.1 Notes Antiderivatives and Indefinite Integrals Antiderivative - derivative backwards - function F is called an antiderivative of the function f if F’(x) = f(x ) EX hFind F ( x ) of f ( x ) FCy ) X v( = 2x — xJxy^- AVPc de X)% . ' 1 ^AX X aisb be C bean O' OUHVI -W A/ ( -< S or y? c~ V/4u rf - i Qi C Antiderivatives of f - If F is an antiderivative of f, then all the antiderivatives of f are of the form F(x) + C, where C is a constant EX 2 : Find the antiderivative of f { x ) = x 5 X 0> r -c FOO '( x) + = y* y RULE: All the antiderivatives of f ( x ) = LA/M^-£ P I k\ \ F( x ) = t ( EX 3: Find the antiderivative of f ( x ) 'X c. = x —I 2 3u c -f . J3 xn are of the form: $\Cy # IS C U\Aj CkC< p\r c is CK. c _ p~ +- £ 3* - ' V'" Indefinite Integrals ( / ) - Let F be an antiderivative of the function f. The indefinite integral of f, denoted by f f ( x ) dx , is defined as f { x ) dx = F (x) + C , where C is a constant Indefinite Integral = all the antiderivatives of f o fX /A ft IY\ ( of rr x -A > S { v kx Vc (( oryy^c[ I f { x )dx tz f O0> - ~ - fo be . ft I ff ^ re -Vpr c-f -b 'V. KVW.o( • n~ f Basic Integration Rules J I / I xn dx = [ f i x ) + # (x ) ]dx [ f i x ) - g i x )]d x J = cx + C c dx X +C n+1 f ( x )d x + J =J = n+1 f ( x )d x Jex c f i x )d x =c I / J — exdx - J J g ( x )d x g i x )d x f i x )d x +C 1 eaxdx = - eax + C a EX 4: / 5 dx -dx = ln|x| + C EX 5: / 3x 4 dx 5 y HC 3% &~ +- c * * /4 X -0 S 4 EX 7: / (x 2 + x 3 ) d x EXJi: / x 6 dx •4 . XI 4 XZ 3 M 4- C c( 3 X* -4- H " r " ^ EX 8: J ( 7x 5 + x 2 7xu * — x ) dx 2. X3 4? L-fe l i < — r / ^ X X — dx - dy v ) ^ ^ X ' / J X5 / — 3 3@ f. K. + - c. _ l, / ( 2 ) = = i r x»- x v c — -/ £ X. 3 C =- - c -f - c. x ' >} c -W -K (3) -bc. -SxM =3 EX 19 : Solve the differential equation. f ' ( x ) = x 2 2 x + 2 , / ( 3) ) 2, QX '3O +- 2-'X 1 + 3 3 st pc* = \ ai3 -t XCoid \ CJ V r(h ~ * EX 17: f d 2 - 2 c s c 20 d d << • zzrS f/ n ' < 0 i W C y - fap' X EX 18 : Solve the differential equation. f { x ) = 3 x ~ _ X- ' P c 1 C * = —1 J-'X - C ic