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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the function\[\int f'(ax+b)[f(ax+b)]^n\]

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Toolbox:
  • If an integral function $f(x)=t,$ then $f'(x)dx=dt$ then $\int f(x)dx=\int tdt$
  • (ii)$\int x^n dx=\frac{x^{n+1}}{n+1}+c$
Given $I=\int f'(ax+b)[f(ax+b)]^n$
 
Let $f(ax+b)=t$
 
On differentiating with respect to x
 
$f'(ax+b)dx=dt \qquad =>f'(ax+b)dx=\frac{dt}{a}$
 
Substituting for t and dt
 
Therefore $I=\frac{1}{a}\int t^n.dt$
 
On integrating we get
 
$\frac{1}{a} \bigg[\frac{t^{n+1}}{n+1}+c\bigg]$
 
Substituting for t we get
 
$I=\frac{1}{a}[f(ax+b)]^{n+1} \times \frac{1}{n+1}+c$
 
$=\frac{1}{a(n+1)}[f(ax+b)]^{n+1}+c$

 

 

answered Feb 17, 2013 by meena.p
 
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