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: ![1 over 20 log invisible function application 3](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
. We know that there are some integrals where we can use substitution method which makes problem to solve easily. We will use one of them.
We will use the substitution method.
So as per the question, we have:
![integral subscript 0 superscript pi divided by 4 end superscript fraction numerator sin invisible function application x plus cos invisible function application x over denominator 9 plus 16 sin invisible function application 2 x end fraction d x
N o w space l e t comma space sin x minus cos x equals t space space........................ e q. space 1
D i f f e r e n t i a t i n g space i t space w. r. t. x comma space w e space g e t colon
left parenthesis cos x plus sin x right parenthesis d x equals d t
S q u a r i n g space e q.1 comma space w e space g e t colon
sin squared x plus cos squared x minus 2 sin x cos x equals t squared
1 minus sin 2 x equals t squared
sin 2 x equals 1 minus t squared
N o w space c h a n g i n g space t h e space l i m i t s comma space w e space g e t colon
W h e n space x equals 0 rightwards double arrow t equals sin 0 minus cos 0 equals negative 1
W h e n space x equals straight pi over 4 rightwards double arrow t equals sin straight pi over 4 minus cos straight pi over 4 equals 0
N o w space p u t t i n g space t h i s space i n space o r i g i n a l space i n t e g r a l comma space w e space g e t colon
integral subscript 0 superscript pi divided by 4 end superscript fraction numerator sin invisible function application x plus cos invisible function application x over denominator 9 plus 16 sin invisible function application 2 x end fraction d x equals integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator 9 plus 16 left parenthesis 1 minus t squared right parenthesis end fraction
integral subscript 0 superscript pi divided by 4 end superscript fraction numerator sin invisible function application x plus cos invisible function application x over denominator 9 plus 16 sin invisible function application 2 x end fraction d x equals integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator 9 plus 16 minus 16 t squared end fraction
integral subscript 0 superscript pi divided by 4 end superscript fraction numerator sin invisible function application x plus cos invisible function application x over denominator 9 plus 16 sin invisible function application 2 x end fraction d x equals integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator 25 minus 16 t squared end fraction
integral subscript 0 superscript pi divided by 4 end superscript fraction numerator sin invisible function application x plus cos invisible function application x over denominator 9 plus 16 sin invisible function application 2 x end fraction d x equals 1 over 16 integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator begin display style 25 over 16 end style minus t squared end fraction
1 over 16 integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator begin display style 25 over 16 minus t squared end style end fraction equals 1 over 16 open square brackets fraction numerator 1 over denominator 2 cross times begin display style 5 over 4 end style end fraction cross times log open vertical bar fraction numerator begin display style 5 over 4 end style plus t over denominator begin display style 5 over 4 end style minus t end fraction close vertical bar close square brackets subscript negative 1 end subscript superscript 0
1 over 16 integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator begin display style 25 over 16 minus t squared end style end fraction equals 1 fourth open square brackets 1 over 10 cross times log open vertical bar fraction numerator begin display style 5 plus 4 t end style over denominator begin display style 5 minus 4 end style end fraction close vertical bar close square brackets subscript negative 1 end subscript superscript 0
1 over 16 integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator begin display style 25 over 16 minus t squared end style end fraction equals 1 fourth cross times 1 over 10 cross times open square brackets log open vertical bar fraction numerator begin display style 5 plus 0 end style over denominator begin display style 5 minus 0 end style end fraction close vertical bar minus log open vertical bar fraction numerator begin display style 5 minus 4 end style over denominator begin display style 5 plus 4 end style end fraction close vertical bar close square brackets
1 over 16 integral subscript negative 1 end subscript superscript 0 fraction numerator d t over denominator begin display style 25 over 16 minus t squared end style end fraction equals 1 over 40 cross times open square brackets log space 1 plus log space 9 close square brackets
equals fraction numerator log space 9 over denominator 40 end fraction equals fraction numerator 2 cross times log space 3 over denominator 40 end fraction equals fraction numerator log space 3 over denominator 20 end fraction](data:image/png;base64,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)
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
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 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