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- Improper integral
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How to use it?
- Integral of d{x}:
- Integral of sin
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Integral of ln(cosx)/cos^2x
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Integral of 2x^2-1
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Integral of sin5xcosx
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Identical expressions
- ln(cosx)/cos^2x
- ln( co sinus of e of x) divide by co sinus of e of squared x
- ln(cosx)/cos2x
- lncosx/cos2x
- ln(cosx)/cos²x
- ln(cosx)/cos to the power of 2x
- lncosx/cos^2x
- ln(cosx) divide by cos^2x
- ln(cosx)/cos^2xdx
Integral of ln(cosx)/cos^2x dx
The solution
You have entered
[src]
1 / | | log(cos(x)) | ----------- dx | 2 | cos (x) | / 0
$$\int\limits_{0}^{1} \frac{\log{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}}\, dx$$
The answer (Indefinite)
[src]
/ 2/x\ \
| tan |-| |
| 1 \2/ | /x\
2*log|----------- - -----------|*tan|-|
/ /x\ 2/x\ | 2/x\ 2/x\| \2/
| 2*tan|-| x*tan |-| |1 + tan |-| 1 + tan |-||
| log(cos(x)) x \2/ \2/ \ \2/ \2//
| ----------- dx = C + ------------ - ------------ - ------------ - ---------------------------------------
| 2 2/x\ 2/x\ 2/x\ 2/x\
| cos (x) -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-|
| \2/ \2/ \2/ \2/
/
$$2\,\left(-{{\sin x\,\log \left({{1-{{\sin ^2x}\over{\left(\cos x+1
\right)^2}}}\over{{{\sin ^2x}\over{\left(\cos x+1\right)^2}}+1}}
\right)}\over{\left(\cos x+1\right)\,\left({{\sin ^2x}\over{\left(
\cos x+1\right)^2}}-1\right)}}-\arctan \left({{\sin x}\over{\cos x+1
}}\right)-{{\sin x}\over{\left(\cos x+1\right)\,\left({{\sin ^2x
}\over{\left(\cos x+1\right)^2}}-1\right)}}\right)$$
The graph
The answer
[src]
/ 2 \
| 1 tan (1/2) |
2*log|------------- - -------------|*tan(1/2)
2 | 2 2 |
1 tan (1/2) 2*tan(1/2) \1 + tan (1/2) 1 + tan (1/2)/
-------------- - -------------- - -------------- - ---------------------------------------------
2 2 2 2
-1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2)
$$-2\,\arctan \left({{\sin 1}\over{\cos 1+1}}\right)+{{\sin 1\,\log
\cos 1}\over{\cos 1}}+{{\sin 1}\over{\cos 1}}$$
=
=
/ 2 \
| 1 tan (1/2) |
2*log|------------- - -------------|*tan(1/2)
2 | 2 2 |
1 tan (1/2) 2*tan(1/2) \1 + tan (1/2) 1 + tan (1/2)/
-------------- - -------------- - -------------- - ---------------------------------------------
2 2 2 2
-1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2)
$$\frac{1}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{2 \log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{2 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$
The graph
Use the examples entering the upper and lower limits of integration.