Mister Exam
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Integral of 1/
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Integral of dy/y^2
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Integral of x/x
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Integral of x/cos^2x
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Identical expressions
- x/cos^2x
- x divide by co sinus of e of squared x
- x/cos2x
- x/cos²x
- x/cos to the power of 2x
- x divide by cos^2x
- x/cos^2xdx
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Similar expressions
- tan^2(x)/cos^2(x)
- sinxdx/cos^2*x
- (7-3sin^2x)/(cos^2x)
- (sqrt(1-tanx))/(cos^2x)
Integral of x/cos^2x dx
The solution
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[src]
1 / | | x | ------- dx | 2 | cos (x) | / 0
$$\int\limits_{0}^{1} \frac{x}{\cos^{2}{\left(x \right)}}\, dx$$
Integral(x/cos(x)^2, (x, 0, 1))
The answer (Indefinite)
[src]
/ / 2/x\\ / /x\\ / /x\\ 2/x\ / /x\\ 2/x\ / /x\\ 2/x\ / 2/x\\ /x\ | log|1 + tan |-|| log|1 + tan|-|| log|-1 + tan|-|| tan |-|*log|1 + tan|-|| tan |-|*log|-1 + tan|-|| tan |-|*log|1 + tan |-|| 2*x*tan|-| | x \ \2// \ \2// \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ | ------- dx = C + ---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------ | 2 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ | cos (x) -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| | \2/ \2/ \2/ \2/ \2/ \2/ \2/ /
$$\int \frac{x}{\cos^{2}{\left(x \right)}}\, dx = C - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$
The graph
The answer
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/ 2 \ 2 2 2 / 2 \
log\1 + tan (1/2)/ pi*I + log(1 - tan(1/2)) log(1 + tan(1/2)) 2*tan(1/2) tan (1/2)*(pi*I + log(1 - tan(1/2))) tan (1/2)*log(1 + tan(1/2)) tan (1/2)*log\1 + tan (1/2)/
------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
2 2 2 2 2 2 2
-1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2)
$$\frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{2 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$
=
=
/ 2 \ 2 2 2 / 2 \
log\1 + tan (1/2)/ pi*I + log(1 - tan(1/2)) log(1 + tan(1/2)) 2*tan(1/2) tan (1/2)*(pi*I + log(1 - tan(1/2))) tan (1/2)*log(1 + tan(1/2)) tan (1/2)*log\1 + tan (1/2)/
------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
2 2 2 2 2 2 2
-1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2)
$$\frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{2 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$
log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2) - pi*i - (pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) - log(1 + tan(1/2))/(-1 + tan(1/2)^2) - 2*tan(1/2)/(-1 + tan(1/2)^2) + tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) + tan(1/2)^2*log(1 + tan(1/2))/(-1 + tan(1/2)^2) - tan(1/2)^2*log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2)
The graph
Use the examples entering the upper and lower limits of integration.