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Question: Integrate and simplify the following expression \(\int {{\text{sin (ax + }}{\text{ b)cos (ax + }}{\t...

Integrate and simplify the following expression ∫sin (ax +  b)cos (ax +  b)dx\int {{\text{sin (ax + }}{\text{ b)cos (ax + }}{\text{ b)dx}}}

Explanation

Solution

Hint: To solve this question, we will use the substitution method. We will substitute the value of cos⁡ (ax + b)\cos {\text{ }}({\text{ax + }}{\text{b)}}so that the integration becomes simplified and we can easily solve it.

Complete step-by-step answer:

Now, we will use the substitution method. Substitution method is used to make the given integral in a simplified form so that it can be integrated with the use of one or few properties. Now,
Let cos⁡ (ax + b) = t\cos {\text{ }}({\text{ax + b) = t}}
As, d (cos ax)dx = - a(sin ax) \dfrac{{{\text{d (cos ax)}}}}{{{\text{dx}}}}{\text{ = - a(sin ax) }}
Differentiating both sides with respect to x, we get
 - a sin(ax + b)dx = dt{\text{ - a sin(ax + b)dx = dt}}
sin(ax + b)dx = - dta{\text{sin(ax + b)dx = - }}\dfrac{{{\text{dt}}}}{{\text{a}}}
Now, I = ∫sin (ax +  b)cos (ax +  b)dx\int {{\text{sin (ax + }}{\text{ b)cos (ax + }}{\text{ b)dx}}}
I = −1a∫tdt- \dfrac{1}{{\text{a}}}\int {{\text{tdt}}}
Now, ∫xndx = xn + 1n + 1\int {{{\text{x}}^{\text{n}}}{\text{dx}}} {\text{ = }}\dfrac{{{{\text{x}}^{{\text{n + 1}}}}}}{{{\text{n + 1}}}}
Therefore, I =  - t22a + C{\text{ - }}\dfrac{{{{\text{t}}^2}}}{{{\text{2a}}}}{\text{ + C}}
Now, putting cos⁡ (ax + b) = t\cos {\text{ }}({\text{ax + b) = t}}, we get
I = −cos2(ax +  b)2a + C - \dfrac{{{\text{co}}{{\text{s}}^2}({\text{ax + }}{\text{ b)}}}}{{2{\text{a}}}}{\text{ + C}}, where C is the integration constant.
Therefore, ∫sin (ax +  b)cos (ax +  b)dx\int {{\text{sin (ax + }}{\text{ b)cos (ax + }}{\text{ b)dx}}} = −cos2(ax +  b)2a + C - \dfrac{{{\text{co}}{{\text{s}}^2}({\text{ax + }}{\text{ b)}}}}{{2{\text{a}}}}{\text{ + C}}

Note: When we come up with such types of questions, when we use substitution method and let any function equal to a variable, it is important that after finding the value of integration, you should replace the variable with the function and write the final answer in the terms of that function as in the above question, we let cos⁡ (ax + b) = t\cos {\text{ }}({\text{ax + b) = t}} and solve the question in terms of t, but in the final answer, we put the value of t and write the value of integral in terms of cos⁡ (ax + b)\cos {\text{ }}({\text{ax + }}{\text{b)}}. This question can be solved by using the property sin2x = 2 sin xcos x. We can apply this property in the integral and find the integral. This is also the easiest method to solve the given question.