Question
- -1
- 2
![1 plus e to the power of negative 1 end exponent](data:image/png;base64,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)
![2 minus 1 over straight e](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC0AAAAjCAYAAAAAEIPqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAPxJREFUeNpjYBg8QA2Ia4H4AsMQAouBOA2I/zMMQTDq6FFHjzp61NGjjh4mjrYG4jVA/AmIf0Gr0ugBcCw6xgsOAnEkEPNA+VpAfBQqNqSAPBBfGorp68dQc7AlNIkMGcABxCehGXRIAEEg3gDEbmTkdmJz/38yMVagBHWwylAJYQ0gng3EXEPFweJAvAqIWYZSxtsCDekhV9+TmplGwSgYBfgBqEX5GNremQ+tlQc1MAfiG0AsC8RMDJCRpr7B7uh1WNruzwe7o79AKzhk8Gmo1hODGrwbSm0eGACVFslDzdGgpvBdpDY8qH+6cCg43BTavfsDxFeA2A+XQgAQBlQVYS6vzAAAAJJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW4+MjwvbW4+PG1vPiYjeDIyMTI7PC9tbz48bWZyYWM+PG1uPjE8L21uPjxtaSBtYXRodmFyaWFudD0ibm9ybWFsIj5lPC9taT48L21mcmFjPjwvbWF0aD4UgwlSAAAAAElFTkSuQmCC)
Hint:
Integrate the each term from the given range.
The correct answer is: ![2 minus 1 over straight e](data:image/png;base64,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)
Given That:
![integral subscript 0 superscript 1 open parentheses 1 plus e to the power of negative x end exponent close parentheses d x equals](data:image/png;base64,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)
>>> Integrate each term:
= ![integral subscript 0 superscript 1 left parenthesis 1 plus e to the power of negative x end exponent right parenthesis d x](data:image/png;base64,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)
>>>Integration of 1 becomes x and integration of e-x becomes -e-x.
>>> = ![left parenthesis 1 minus 1 over e plus 1 right parenthesis](data:image/png;base64,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)
= (2-
)
>>> Therefore, the integration of
becomes (2-
).
=
>>>Integration of 1 becomes x and integration of e-x becomes -e-x.
>>> =
= (2-)
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 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