Question
Hint:
Integrate the every term of the given expression.
The correct answer is: ![fraction numerator 1 over denominator log invisible function application a end fraction minus fraction numerator 1 over denominator log invisible function application b end fraction](data:image/png;base64,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)
Given That:
![integral subscript 0 superscript straight infinity open parentheses straight a to the power of negative straight x end exponent minus straight b to the power of negative straight x end exponent close parentheses dx equals left parenthesis straight a greater than 1 comma straight b greater than 1 right parenthesis](data:image/png;base64,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)
>>> Integrating every term gives :
= ![integral subscript 0 superscript straight infinity open parentheses straight a to the power of negative straight x end exponent minus straight b to the power of negative straight x end exponent close parentheses dx equals left parenthesis straight a greater than 1 comma straight b greater than 1 right parenthesis](data:image/png;base64,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)
= ![integral subscript 0 superscript infinity left parenthesis a to the power of negative x end exponent right parenthesis space minus space integral subscript 0 superscript infinity left parenthesis b to the power of negative x end exponent right parenthesis](data:image/png;base64,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)
=
.
>>> Therefore, the integration of
is
.
Therefore, the integration of is
.
Related Questions to study
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