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Question: What is the integral of \(\int{\left[ x.\cos \left( {{x}^{2}} \right) \right]\left( dx \right)?}\)...

What is the integral of ∫[x.cos⁡(x2)](dx)?\int{\left[ x.\cos \left( {{x}^{2}} \right) \right]\left( dx \right)?}

Explanation

Solution

We solve this question by using the substitution method of solving integrals. We substitute the value of x2{{x}^{2}} as a variable t. We then find the value of dxdx and substitute for it in the integral. Then we simplify using the basic integration formula to solve the given integral. After simplifying and obtaining the answer, we substitute back for t as x2.{{x}^{2}}.

Complete step by step solution:
In order to solve this question, let us consider the given integral in the question.
=∫[x.cos⁡(x2)](dx)= \int{\left[ x.\cos \left( {{x}^{2}} \right) \right]\left( dx \right)}
Let us consider the value x2{{x}^{2}} as a variable t. Let us assume,
=t=x2…(1)= t={{x}^{2}}\ldots \left( 1 \right)
We differentiate this above equation and we know the differentiation of x2{{x}^{2}} with respect to x is 2x.
=dt=2xdx= dt=2xdx
We divide both sides of the equation by 2.
=12dt=xdx= \dfrac{1}{2}dt=xdx
We substitute the value of x2{{x}^{2}} as t and the value of xdxxdx by 12dt\dfrac{1}{2}dt in the integral. Before that, we rearrange the terms,
=∫[cos⁡(x2)]x.dx= \int{\left[ \cos \left( {{x}^{2}} \right) \right]x.dx}
Now, we make the above said substitutions,
=∫cos⁡(t).12dt= \int{\cos \left( t \right).\dfrac{1}{2}dt}
Taking the constant 12\dfrac{1}{2} outside the integral,
=12∫cos⁡(t)dt= \dfrac{1}{2}\int{\cos \left( t \right)dt}
We know the integral of a cos⁡t\cos t is given by sin⁡t\sin t and we add a constant of integration after it.
=12∫cos⁡(t)dt=12sin⁡t+C= \dfrac{1}{2}\int{\cos \left( t \right)dt}=\dfrac{1}{2}\sin t+C
Now, we substitute back for the variable t in terms of x given by the equation 1.
=12sin⁡(x2)+C= \dfrac{1}{2}\sin \left( {{x}^{2}} \right)+C

Hence, the integral of the above function is found to be 12sin⁡(x2)+C,\dfrac{1}{2}\sin \left( {{x}^{2}} \right)+C, where C is the constant of integration.

Note:
We need to know the basic formulae for the integration of basic functions. We can also solve this question by taking integration by parts for the two terms xx and cos⁡(x2)\cos \left( {{x}^{2}} \right) and simplifying it using the basic integration formulae. It is important to know the integration formula for trigonometric functions to solve this question.