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How to use it?
- Integral of d{x}:
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Integral of e^3
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Integral of exp(4*x)
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Integral of 1/√(4-x^2)
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Integral of 3^x*e^x
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Identical expressions
- (sin(x))/(one +sin(x)-cos(x))
- ( sinus of (x)) divide by (1 plus sinus of (x) minus co sinus of e of (x))
- ( sinus of (x)) divide by (one plus sinus of (x) minus co sinus of e of (x))
- sinx/1+sinx-cosx
- (sin(x)) divide by (1+sin(x)-cos(x))
- (sin(x))/(1+sin(x)-cos(x))dx
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Similar expressions
- (sin(x))/(1+sin(x)+cos(x))
- (sin(x))/(1-sin(x)-cos(x))
- (sinx)/(1+sinx-cosx)
Integral of (sin(x))/(1+sin(x)-cos(x)) dx
The solution
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pi -- 8 / | | sin(x) | ------------------- dx | 1 + sin(x) - cos(x) | / pi -- 12
$$\int\limits_{\frac{\pi}{12}}^{\frac{\pi}{8}} \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) - \cos{\left(x \right)}}\, dx$$
Integral(sin(x)/(1 + sin(x) - cos(x)), (x, pi/12, pi/8))
Detail solution
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Rewrite the integrand:
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The integral of a constant times a function is the constant times the integral of the function:
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Don't know the steps in finding this integral.
But the integral is
So, the result is:
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Now simplify:
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Add the constant of integration:
The answer is:
The answer (Indefinite)
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/ / 2/x\\ | log|1 + tan |-|| | sin(x) x \ \2// / /x\\ | ------------------- dx = C + - - ---------------- + log|1 + tan|-|| | 1 + sin(x) - cos(x) 2 2 \ \2// | /
$$\int \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) - \cos{\left(x \right)}}\, dx = C + \frac{x}{2} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2}$$
The graph
The answer
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/ 2\
| / ___________\ |
/ 2\ | | / ___ | |
| / ___ ___ \ | | | / 1 \/ 2 | |
| | \/ 2 \/ 6 | | | | / - + ----- | |
| | ----- + ----- | | | | 1 \/ 2 4 | |
| | 1 4 4 | | log|1 + |---------------- - ----------------| |
log|1 + |--------------- - ---------------| | | | ___________ ___________| | / ___________\
| | ___ ___ ___ ___| | / ___ ___ \ | | / ___ / ___ | | | / ___ |
| | \/ 2 \/ 6 \/ 2 \/ 6 | | | \/ 2 \/ 6 | | | / 1 \/ 2 / 1 \/ 2 | | | / 1 \/ 2 |
| |- ----- + ----- - ----- + -----| | | ----- + ----- | | | / - - ----- / - - ----- | | | / - + ----- |
\ \ 4 4 4 4 / / | 1 4 4 | \ \\/ 2 4 \/ 2 4 / / pi | 1 \/ 2 4 |
--------------------------------------------- - log|1 + --------------- - ---------------| - ----------------------------------------------- + -- + log|1 + ---------------- - ----------------|
2 | ___ ___ ___ ___| 2 48 | ___________ ___________|
| \/ 2 \/ 6 \/ 2 \/ 6 | | / ___ / ___ |
| - ----- + ----- - ----- + -----| | / 1 \/ 2 / 1 \/ 2 |
\ 4 4 4 4 / | / - - ----- / - - ----- |
\ \/ 2 4 \/ 2 4 /
$$- \log{\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + 1 + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} \right)} - \frac{\log{\left(\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}\right)^{2} + 1 \right)}}{2} + \frac{\log{\left(\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2} + 1 \right)}}{2} + \frac{\pi}{48} + \log{\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + 1 + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} \right)}$$
=
=
/ 2\
| / ___________\ |
/ 2\ | | / ___ | |
| / ___ ___ \ | | | / 1 \/ 2 | |
| | \/ 2 \/ 6 | | | | / - + ----- | |
| | ----- + ----- | | | | 1 \/ 2 4 | |
| | 1 4 4 | | log|1 + |---------------- - ----------------| |
log|1 + |--------------- - ---------------| | | | ___________ ___________| | / ___________\
| | ___ ___ ___ ___| | / ___ ___ \ | | / ___ / ___ | | | / ___ |
| | \/ 2 \/ 6 \/ 2 \/ 6 | | | \/ 2 \/ 6 | | | / 1 \/ 2 / 1 \/ 2 | | | / 1 \/ 2 |
| |- ----- + ----- - ----- + -----| | | ----- + ----- | | | / - - ----- / - - ----- | | | / - + ----- |
\ \ 4 4 4 4 / / | 1 4 4 | \ \\/ 2 4 \/ 2 4 / / pi | 1 \/ 2 4 |
--------------------------------------------- - log|1 + --------------- - ---------------| - ----------------------------------------------- + -- + log|1 + ---------------- - ----------------|
2 | ___ ___ ___ ___| 2 48 | ___________ ___________|
| \/ 2 \/ 6 \/ 2 \/ 6 | | / ___ / ___ |
| - ----- + ----- - ----- + -----| | / 1 \/ 2 / 1 \/ 2 |
\ 4 4 4 4 / | / - - ----- / - - ----- |
\ \/ 2 4 \/ 2 4 /
$$- \log{\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + 1 + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} \right)} - \frac{\log{\left(\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}\right)^{2} + 1 \right)}}{2} + \frac{\log{\left(\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2} + 1 \right)}}{2} + \frac{\pi}{48} + \log{\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + 1 + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} \right)}$$
log(1 + (1/(-sqrt(2)/4 + sqrt(6)/4) - (sqrt(2)/4 + sqrt(6)/4)/(-sqrt(2)/4 + sqrt(6)/4))^2)/2 - log(1 + 1/(-sqrt(2)/4 + sqrt(6)/4) - (sqrt(2)/4 + sqrt(6)/4)/(-sqrt(2)/4 + sqrt(6)/4)) - log(1 + (1/sqrt(1/2 - sqrt(2)/4) - sqrt(1/2 + sqrt(2)/4)/sqrt(1/2 - sqrt(2)/4))^2)/2 + pi/48 + log(1 + 1/sqrt(1/2 - sqrt(2)/4) - sqrt(1/2 + sqrt(2)/4)/sqrt(1/2 - sqrt(2)/4))
Use the examples entering the upper and lower limits of integration.