
Evaluate the integral
Answer
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Hint: We are asked to find the integral of the given term. Since this is a complex one we need to first solve the stuff inside the bracket and we need to simplify that. In this case we will use several identities of double angle involving cosine and sine and then after simplifying the matter inside the bracket we will integrate the whole term. We should be aware of all the trigonometric identities.
Complete step-by-step solution:
We have .
The term inside the bracket is:
We can use the following identity in place of :
And in place of , we can use the following formula:
Putting in the fraction and simplifying it, we have:
Now, we use the following important identity:
Using the same we get:
Putting this in the denominator:
So, we can say that:
We obtained this using the formula:
Putting this term inside the bracket we have:
And we know that:
Hence, option is correct.
Note: If the formulae related to the double angle of sine and cosine are not known then this becomes a very difficult integral to operate because the terms inside bracket cannot be solved so easily. Moreover, you should be aware about the integrals of some basic algebraic terms such as the polynomial terms.
Complete step-by-step solution:
We have
The term inside the bracket is:
We can use the following identity in place of
And in place of
Putting in the fraction and simplifying it, we have:
Now, we use the following important identity:
Using the same we get:
Putting this in the denominator:
So, we can say that:
We obtained this using the formula:
Putting this term inside the bracket we have:
And we know that:
Hence, option
Note: If the formulae related to the double angle of sine and cosine are not known then this becomes a very difficult integral to operate because the terms inside bracket cannot be solved so easily. Moreover, you should be aware about the integrals of some basic algebraic terms such as the polynomial terms.
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