Question
Hint:
We are aware that differentiation is the process of discovering a function's derivative and integration is the process of discovering a function's antiderivative. Thus, both processes are the antithesis of one another. Therefore, we can say that differentiation is the process of differentiation and integration is the reverse. The anti-differentiation is another name for the integration.
Here we have given:
and we have to integrate it. We will use the formula to find the answer.
The correct answer is: ![4 pi](data:image/png;base64,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)
Now we have given the function as
. Here the lower limit is 0 and upper limit is 4. We know that there are some integralswhere we use certain formulas which makes problem to solve easily. We will use one of them.
We will use the formula here as:
![integral subscript blank superscript blank square root of a squared minus x squared end root d x equals 1 half x square root of a squared minus x squared end root plus a squared over 2 sin to the power of negative 1 end exponent x over a plus c
P u t t i n g space t h e space v a l u e s comma space w e space g e t colon
integral subscript 0 superscript 4 square root of 4 squared minus x squared end root d x equals open square brackets 1 half x square root of 4 squared minus x squared end root plus 4 squared over 2 sin to the power of negative 1 end exponent x over 4 plus c close square brackets subscript 0 superscript 4
integral subscript 0 superscript 4 square root of 4 squared minus x squared end root d x equals open square brackets 1 half x square root of 16 minus x squared end root plus 16 over 2 sin to the power of negative 1 end exponent x over 4 plus c close square brackets subscript 0 superscript 4
P u t t i n g space t h e space l i m i t s comma space w e space g e t colon
equals 1 half cross times 4 square root of 16 minus 4 squared end root plus 16 over 2 sin to the power of negative 1 end exponent 4 over 4 minus 1 half cross times 0 square root of 16 minus 0 squared end root plus 16 over 2 sin to the power of negative 1 end exponent 0 over 4
equals 0 plus 8 cross times sin to the power of negative 1 end exponent 1 minus 0 plus 8 cross times sin to the power of negative 1 end exponent 0
equals 8 cross times straight pi over 2 plus 8 cross times 0
equals 4 straight pi](data:image/png;base64,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)
So here we used the concept of integrals of special function and simplified it. We can also solve it manually but it will take lot of time to come to a final answer hence we used the formula of that. The integral of the given function is
Related Questions to study
So here we used the concept of integrals and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the substitution method which makes the problem to solve easily. The integral of the given function is
So here we used the concept of integrals and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the substitution method which makes the problem to solve easily. The integral of the given function is
So here we used the concept of integrals and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the substitution method which makes problem to solve easily. The integral of the given function is
So here we used the concept of integrals and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the substitution method which makes problem to solve easily. The integral of the given function is
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to a final answer hence we used the formula. The integral of the given function is.
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to a final answer hence we used the formula. The integral of the given function is.
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the formula. The integral of the given function is
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the formula. The integral of the given function is
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the formula to solve. The integral of the given function is
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the formula to solve. The integral of the given function is
Therefore, the integration of is
.
Therefore, the integration of is
.
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the substitution method to solve. The integral of the given function is
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the substitution method to solve. The integral of the given function is
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the special case and the formula of that. The integral of the given function is .
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the special case and the formula of that. The integral of the given function is .
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the special case and the formula of that. The integral of the given function is .
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to final answer hence we used the special case and the formula of that. The integral of the given function is .
=
>>>Integration of 1 becomes x and integration of e-x becomes -e-x.
>>> =
= (2-)
=
>>>Integration of 1 becomes x and integration of e-x becomes -e-x.
>>> =
= (2-)
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to the final answer hence we will use trigonometric formulas. The integral of the given function is
So here we used the concept of integrals of special functions and simplified it. We can also solve it manually but it will take lot of time to come to the final answer hence we will use trigonometric formulas. The integral of the given function is