
If Then, is equal to:
Answer
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Hint: To solve the question given above, we will first find out the values of by putting n = 4 and n = 6 respectively in the equation given. For the calculation of our main aim will be to convert higher powers of the sine function to the sine functions or cosine functions with single power. Then we will put the values of and then solve.
Complete step-by-step answer:
In the first step, we are going to calculate the value of We know that,
When we put n = 4 in the above equation, we get,
Now, we have to find the value of As we know that the direct integration of does not exist, so we will try to reduce the power. For this, we will write with the help of the identity Thus, we will get,
Now, here we will use a trigonometric identity,
By using the above identity, we will get,
Now, we will use the identity,
Thus, we will get,
Now, here we will use another trigonometric identity:
Putting we will get,
Putting the value of from (ii) to (i), we will get,
Now, here we will use the following integration formulas,
Considering the constants we get,
Now, similarly, we will calculate the value of Thus, we have,
Now, we will use a trigonometric identity here,
Thus, after using this identity, we will get,
Now, we will use the identity,
Thus, we will get,
Now, here we will use two trigonometric identities,
Similarly,
Also, 2 sin A sin B = cos (A – B) – cos (A + B)
Thus, we will get,
Now, we will use some integration formulas here,
Thus, we will get,
Considering the constants we get,
Now, we have to find the value of Thus, we will get,
Taking the constant, we get,
Now, we will use two trigonometric identities here,
Putting the values of sin 4x and sin 6x from (vi) and (vii) to (v), we will get,
Hence, the option (c) is the right answer.
Note: We can also use the reduction formula to solve this question. Reduction formula for integral of the sine function is,
Now, we will put n = 6. Thus, we will get,
Now, we will use sin 2x = 2 sin x cos x. Thus, we will get,
Now, we will use the identity,
Thus, we get,
Complete step-by-step answer:
In the first step, we are going to calculate the value of
When we put n = 4 in the above equation, we get,
Now, we have to find the value of
Now, here we will use a trigonometric identity,
By using the above identity, we will get,
Now, we will use the identity,
Thus, we will get,
Now, here we will use another trigonometric identity:
Putting
Putting the value of
Now, here we will use the following integration formulas,
Considering the constants
Now, similarly, we will calculate the value of
Now, we will use a trigonometric identity here,
Thus, after using this identity, we will get,
Now, we will use the identity,
Thus, we will get,
Now, here we will use two trigonometric identities,
Similarly,
Also, 2 sin A sin B = cos (A – B) – cos (A + B)
Thus, we will get,
Now, we will use some integration formulas here,
Thus, we will get,
Considering the constants
Now, we have to find the value of
Taking the constant,
Now, we will use two trigonometric identities here,
Putting the values of sin 4x and sin 6x from (vi) and (vii) to (v), we will get,
Hence, the option (c) is the right answer.
Note: We can also use the reduction formula to solve this question. Reduction formula for integral of the sine function is,
Now, we will put n = 6. Thus, we will get,
Now, we will use sin 2x = 2 sin x cos x. Thus, we will get,
Now, we will use the identity,
Thus, we get,
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