Integration by Substitution

Integration by Substitution

Integration by Substitution

(1) When integrand is a function i.e.,∫f[?(x)] ?'(x) dx:

Here, we put ?(x)=t, so that ?'(x) dx=dt and in that case the integrand is reduced to ∫f(t) dt.

(2) When integrand is the product of two factors such that one is the derivative of the others i.e., I = ∫f(x)f'(x) dx:

In this case we put f(x)=t and convert it into a standard integral.

(3) Integral of a function of the form f(ax + b):

Here we put ax + b = t and convert it into standard integral. Obviously if ∫f(x) dx = ?(x) then ∫f(ax + b) dx = (frac { 1 }{ a }) ?(ax + b) + c.

(4) If integral of a function of the form (frac { f'(x) }{ f(x) })



(5) If integral of a function of the form ({ [f(x)] }^{ n }f'(x))



(6) If the integral of a function of the form (frac { f'(x) }{ sqrt { f(x) } } )



(7) Standard substitutions

Integration by Substitution Problems with Solutions

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Solution:



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Solution:



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Solution:



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Solution:



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Solution: