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