The value of is
(a)
(b)
(c) 3
(d) None of these
Answer
Verified
117.6k+ views
Hint:
Here, we will use the property of definite integrals to rewrite the given integral as the sum of three integrals. Then we will change the limits for every trigonometric function and simplify the integral. We will then integrate the functions individually, apply the limits and add the terms to find the answer.
Formula Used: The definite integral of a continuous function can be written as the sum of the definite integrals and , where lies in the interval .
Complete step by step solution:
The integral is a definite integral with the upper limit and lower limit 0.
We know that the functions and are continuous functions.
The definite integral of a continuous function can be written as the sum of the definite integrals and , where lies in the interval .
We will use this property of definite integrals to simplify the given integral.
Therefore, we get
Here, and lie in the interval .
Now, we know that goes from 0 to , and
goes from 1 to in the interval .
Thus, is greater than in the interval .
Therefore, the integral
becomes .
Next, we know that goes from to 1, then to 0, and finally to in the interval .
The function goes from to 0, then to , and finally back to
in the interval .
Thus, is greater than in the interval
.
Therefore, the integral becomes .
Finally, we know that goes from to , and then to 0 in the interval .
The function goes from to 0, and then to 1 in the interval .
Thus, is greater than
in the interval .
Therefore, the integral becomes .
Now, we will rewrite the functions in the given integrals.
Therefore, we get
Integrating the functions, we get
Substituting the limits, we get
Substitute the values of the trigonometric ratios, we get
Adding the terms of the expression, we get
Therefore, we get the value of the integral as .
Thus, the correct option is option (b).
Note:
Some common mistakes in this question include using the upper limit in place of the lower limit, and using the property of definite integrals incorrectly. We can also draw a rough graph including the functions and to determine the minimum and maximum points.
Here, we will use the property of definite integrals to rewrite the given integral as the sum of three integrals. Then we will change the limits for every trigonometric function and simplify the integral. We will then integrate the functions individually, apply the limits and add the terms to find the answer.
Formula Used: The definite integral
Complete step by step solution:
The integral
We know that the functions
The definite integral
We will use this property of definite integrals to simplify the given integral.
Therefore, we get
Here,
Now, we know that
Thus,
Therefore, the integral
Next, we know that
The function
Thus,
Therefore, the integral
Finally, we know that
The function
Thus,
Therefore, the integral
Now, we will rewrite the functions in the given integrals.
Therefore, we get
Integrating the functions, we get
Substituting the limits, we get
Substitute the values of the trigonometric ratios, we get
Adding the terms of the expression, we get
Therefore, we get the value of the integral
Thus, the correct option is option (b).
Note:
Some common mistakes in this question include using the upper limit in place of the lower limit, and using the property of definite integrals incorrectly. We can also draw a rough graph including the functions
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