4.1Antiderivatives and

Indefinite Integration

Definition of Antiderivative: A function F is called anantiderivative of the function f if for every x in the domainof f

F’(x) = f(x) so, dy = f(x) dx

Integration is denoted by an integral sign .∫∫ +== CxFdxxfy )()(

Integrand Variable ofIntegration

Constant of Integration

F’(x) also = f(x)(first derivative)

Basic Integration Formulas

Cxxdx

Cxxdx

Cxxdx

Cn

xdxx

Ckxkdx

Cdx

nn

+=

+−=

+=

++

=

+=

=

∫∫∫∫

∫∫

+

tansec

cossin

sincos

1

0

2

1

Cxxdxx

Cxxdx

Cxxdxx

+−=

+−=

+=

∫∫∫

csccotcsc

cotcsc

sectansec

2

Integrate

€

3xdx∫1

x 3∫ dx

x∫ dx

2sin xdx∫1dx∫x + 2( )∫ dx

3x 4 − 5x 2 + x( )∫ dx

Cx

+=23 2

∫ −= dxx 3 Cx

+−

=−

2

2

Cx

+−= 221

dxx21

∫= Cx

+=23

23

Cx

+=3

2 23

∫= xdxsin2 CxCx +−=+−= cos2)cos(2

Cx +=

Cxx

++= 22

2

Cxxx

++−=23

553 235

Rewriting before integrating

dxx

x∫

+1dxxx

x 1+=∫ ( )dxxx∫ −+= 2121

Cxx

++= 2123

23

2

dxx

x∫ 2cossin

€

=1

cos x

⎛

⎝ ⎜

⎞

⎠ ⎟sin x

cos x

⎛

⎝ ⎜

⎞

⎠ ⎟∫ dx dxxx∫= tansec

Cx +=sec

Find the general solution of the equation F’(x) = and

find the particular solution given the point F(1) = 0.

2

1

x

dxx

xF ∫= 2

1)( dxx∫ −= 2

Cx

Cx

+−=+−

=− 11

1

Cx

y +−=∴1

Now plug in (1,0) and solve for C.

0 = -1 + C

C = 1

Final answer.1

1+−=

xy