INTEGRATION

INTEGRATION

Integration is means adding small parts to get the whole.

Using integration area, volume and other useful thinks can be find out.

without integration, physics and engineering problems can be solved

 integration is represented by ∫ which is the stylish form of S where S is sum

Indefinite integral

If the limit of integral is not given then it is Indefinite integral

∫2x²dx=x³+c

where c is constant and called constant of integration.

Definite integral

if the limit of integration is given then it is called definite integral.

Here constant of integration is not used.

Some standard function and integral

Integrate y=∫cos²xdx

y=∫cos²xdx

=∫(1+cos2x)/2dx

=⟦x+(sin2x/2)⟧/2+c

=x/2+sin2x/4+c

 Integrate  y=∫e^2x dx

y=∫e^2x dx

y=e^2x/2+c

c is constant of integration

Integrate y=∫(√1+sin2x)dx

y=∫(√1+sin2x)dx

But 1=sin²x+cos²x

hence

y=∫√(sin²x+cos²x+sin2x)dx

y=∫√(sin²x+cos²x+2sinxcosx)dx   beause sin2x=2sinxcosx

y=∫√(sinx+cosx)²dx     beause x²+y²+2xy=(x+y)²

y=∫(sinx+cosx)dx

y=-cosx+sinx+c

c is constant of integration

Integrate y=∫(cosx+sinx)/(cosx-sinx) dx

y=∫(cosx+sinx)/(cosx-sinx) dx

y=∫(cosx+sinx/cosx-sinx)×(cosx+sinx/cosx+sinx)dx

y=∫(cosx+sinx)²/(cos²x-sin²x)dx

y=∫(cos²x+sin²x+2sinxcosx)/cos2x dx

y=∫(1+sin2x)/cos2x dx

y=∫(sec2x+tan2x)dx

y=ln(sec2x+tan2x)/2+ln(sec2x)/2+c
c is constant of integration

Determine the equation of the function that has a slope of -5 and crosses the x-axis at (3,0).
Given slope i.e. dy/dx=-5

then

dy=-5dx

integrate both sides

∫dy=-∫5dx

y=-5x+c

it passes through (3,0)

i.e.

0=-5×3+c

c=15

y=-5x+15

What is the integral of sqrt (1-sin2x)?
I=∫√(1-sin2x)dx

=∫√(sin²x+cos²x-2sinxcosx)dx

=∫√(sinx-cosx)²dx

=∫(sinx-cosx)dx

I=-cosx-sinx

solve y=∫dx/(e^x+e^-x)

=∫dxe^x/(e^x+e^-x)e^x

⁼∫dxe^x/(e^2x+1)

put e^x=t

e^xdx=dt

=dt/(t²+1)

=arctant

=arctan(e^x)