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Evaluate the given Integral:

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Hint: Integral expression contains a function and its derivative, hence we use substitution method to reduce the integral into standard form.

Consider the expression,
It consists of two functions, where one function i.e. cosx is the derivative of another which is sinx.
So, we can use the method of integration by substitution.
In this method we substitute one of the functions to reduce the expression into standard form.
Now let us consider, cosx=t
Differentiating both sides with respect to x, we get
We know,d(cosx)dx=sinx
d(cosx)dx =dtdx 
Substitute sinxdx=dt in the expression, we get
 (cosx)sinxdx=(t )dt
We know, xndx= xn+1n+1+C

So, after integrating we get
 (t )dt=t3232+C
 (t )dt=23 t32+C
Re-substitute the value of t in terms of x, we get
 (cosx)sinxdx=23 cos32x+C
Note: Whenever an integrating expression consists of more than one function convert it into standard form by reduction method of integration such as substitution method of integration
Last updated date: 24th Sep 2023
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