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Integration by Substitution


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Overview

Integration by Substitution :

When f(x)=h(g(x))g(x)

then,
f(x)dx=h(y)dy
where y=g(x)
and dy=g(x)dx and so
 »  f(x)dx=[h(y)dy]y=g(x)+c

substitute

Consider the integration 2xsinx2dx
We notice that ddxx2=2x is part of the integration.

t=x2 and substitute in the integrand.

integration 2xsinx2dx

Substitute t=x2 and dt=2xdx.

2xsinx2dx

=sintdt

=-cost+c

substitute t=x2
=-cosx2+c

Considering the integration f(x)dx :
If it is observed that f(x) can be rewritten as
f(x)=h[g(x)]g(x)

then,
f(x)dx=h(y)dy
where y=g(x)
and dy=g(x)dx

This method is "integration by substitution".

summary

Integration by Substitution: When f(x)=h(g(x))g(x)

then,
f(x)dx=h(y)dy
where y=g(x)
and dy=g(x)dx and so f(x)dx=[h(y)dy]y=g(x)+c

parametric form

Given a function in parametric form y=rcost and x=rsint, To proceed with the integration, substitute y=rcost and dx=d(rcost). Then, proceed with variable of integration as t

summary

Given a function in parametric form y=rcost and x=rsint.
ydx

substituting y=rcost and dx=d(rsint). the d() means derivative of
=rcostd(rsint)

=rcostrcostdt

=r2cos2tdt

This can be integrated.

tan

Integrate tanxdx

This can be integrated with tanx=-(cosx)/cosx

tanxdx

=-(cosx)cosx

=-log|cosx|+c

cot

Integrate cotxdx

This can be integrated with cotx=(sinx)/sinx

cotxdx

=(sinx)sinx

=log|sinx|+c

sec

Integrate secxdx

This can be integrated with secx=(secx+tanx)/(secx+tanx)

secxdx

=(secx+tanx)secx+tanxdx

=log|secx+tanx|+c

cosec

Integrate cscxdx

This can be integrated with cscx=-(cscx+cotx)/(cscx+cotx)

cscxdx

=-(cscx+cotx)cscx+cotxdx

=-log|cscx+cotx|+c

summary

Integration by Substitution

When f(x)=h(g(x))g(x)

then,
f(x)dx=h(y)dy
where y=g(x)
and dy=g(x)dx and so
 »  f(x)dx=[h(y)dy]y=g(x)+c

Outline

The outline of material to learn "Integral Calculus" is as follows.

•   Detailed outline of Integral Calculus

    →   Application Scenario

    →   Integration First Principles

    →   Graphical Meaning of Integration

    →   Definition of Integrals

    →   Fundamental Theorem of Calculus

    →   Algebra of Integrals

    →   Antiderivatives: Standard results

    →   Integration of Expressions

    →   Integration by Substitution

    →   Integration using Identities

    →   Integration by Parts

    →   Integration by Partial Fraction

    →   Integration: Combination of Methods