Question

Evaluate the given Integral:
∫(cos⁡x)sinxdx

Answer 1

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

Consider the expression,
∫(cos⁡x)sinxdx
It consists of two functions, where one function i.e. cos⁡x is the derivative of another which is sin⁡x.
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, cos⁡x=t
Differentiating both sides with respect to x, we get
We know,d(cos⁡x)dx=−sin⁡x
Now,
d(cos⁡x)dx =dtdx
−sin⁡x=dtdx
−sin⁡xdx=dt
Substitute sin⁡xdx=−dt in the expression, we get
∫(cos⁡x)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
∫(cos⁡x)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