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Identical expressions
- one /cos^ two (kx+b)
- 1 divide by co sinus of e of squared (kx plus b)
- one divide by co sinus of e of to the power of two (kx plus b)
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Similar expressions
- 1/cos^2(kx-b)
Integral of 1/cos^2(kx+b) dx
The solution
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[src]
1 / | | 1 | ------------- dx | 2 | cos (k*x + b) | / 0
$$\int\limits_{0}^{1} \frac{1}{\cos^{2}{\left(b + k x \right)}}\, dx$$
Integral(1/(cos(k*x + b)^2), (x, 0, 1))
The answer (Indefinite)
[src]
// / / -pi pi \ / pi \\\
|| zoo*x for And|Or|b = ----, b = - -- - k*x|, Or|b = - -- - k*x, k = 0|||
|| \ \ 2 2 / \ 2 //|
|| |
|| x |
/ || ------- for k = 0 |
| || 2 |
| 1 || cos (b) |
| ------------- dx = C + |< |
| 2 || /b k*x\ |
| cos (k*x + b) || -2*tan|- + ---| |
| || \2 2 / |
/ ||-------------------- otherwise |
|| 2/b k*x\ |
||-k + k*tan |- + ---| |
|| \2 2 / |
\\ /
$$\int \frac{1}{\cos^{2}{\left(b + k x \right)}}\, dx = C + \begin{cases} \tilde{\infty} x & \text{for}\: \left(b = - \frac{\pi}{2} \vee b = - k x - \frac{\pi}{2}\right) \wedge \left(b = - k x - \frac{\pi}{2} \vee k = 0\right) \\\frac{x}{\cos^{2}{\left(b \right)}} & \text{for}\: k = 0 \\- \frac{2 \tan{\left(\frac{b}{2} + \frac{k x}{2} \right)}}{k \tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} - k} & \text{otherwise} \end{cases}$$
The answer
[src]
/ / -pi \ | nan for And|b = ----, k = 0| | \ 2 / | | 1 | / | | | | / pi | | | 0 for b = - -- - k*x | | | 2 | | | | | < 1 dx for k = 0 | | |------- otherwise | | | 2 | | |cos (b) | | \ | | | / | 0 < | 1 | / | | | | / pi | | | 0 for b = - -- - k*x | | | 2 | | | | | | / 2/b k*x\\ 2 2/b k*x\ / 2/b k*x\\ | | | k*|1 + tan |- + ---|| 2*k *tan |- + ---|*|1 + tan |- + ---|| | | < \ \2 2 // \2 2 / \ \2 2 // dx otherwise | | |- --------------------- + -------------------------------------- otherwise | | | 2/b k*x\ 2 | | | -k + k*tan |- + ---| / 2/b k*x\\ | | | \2 2 / |-k + k*tan |- + ---|| | | | \ \2 2 // | | \ | | |/ \0
$$\begin{cases} \text{NaN} & \text{for}\: b = - \frac{\pi}{2} \wedge k = 0 \\\int\limits_{0}^{1} \begin{cases} 0 & \text{for}\: b = - k x - \frac{\pi}{2} \\\frac{1}{\cos^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\, dx & \text{for}\: k = 0 \\\int\limits_{0}^{1} \begin{cases} 0 & \text{for}\: b = - k x - \frac{\pi}{2} \\\frac{2 k^{2} \left(\tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)}}{\left(k \tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} - k\right)^{2}} - \frac{k \left(\tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} + 1\right)}{k \tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} - k} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
=
=
/ / -pi \ | nan for And|b = ----, k = 0| | \ 2 / | | 1 | / | | | | / pi | | | 0 for b = - -- - k*x | | | 2 | | | | | < 1 dx for k = 0 | | |------- otherwise | | | 2 | | |cos (b) | | \ | | | / | 0 < | 1 | / | | | | / pi | | | 0 for b = - -- - k*x | | | 2 | | | | | | / 2/b k*x\\ 2 2/b k*x\ / 2/b k*x\\ | | | k*|1 + tan |- + ---|| 2*k *tan |- + ---|*|1 + tan |- + ---|| | | < \ \2 2 // \2 2 / \ \2 2 // dx otherwise | | |- --------------------- + -------------------------------------- otherwise | | | 2/b k*x\ 2 | | | -k + k*tan |- + ---| / 2/b k*x\\ | | | \2 2 / |-k + k*tan |- + ---|| | | | \ \2 2 // | | \ | | |/ \0
$$\begin{cases} \text{NaN} & \text{for}\: b = - \frac{\pi}{2} \wedge k = 0 \\\int\limits_{0}^{1} \begin{cases} 0 & \text{for}\: b = - k x - \frac{\pi}{2} \\\frac{1}{\cos^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\, dx & \text{for}\: k = 0 \\\int\limits_{0}^{1} \begin{cases} 0 & \text{for}\: b = - k x - \frac{\pi}{2} \\\frac{2 k^{2} \left(\tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)}}{\left(k \tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} - k\right)^{2}} - \frac{k \left(\tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} + 1\right)}{k \tan^{2}{\left(\frac{b}{2} + \frac{k x}{2} \right)} - k} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((nan, (k = 0)∧(b = -pi/2)), (Integral(Piecewise((0, b = -pi/2 - k*x), (cos(b)^(-2), True)), (x, 0, 1)), k = 0), (Integral(Piecewise((0, b = -pi/2 - k*x), (-k*(1 + tan(b/2 + k*x/2)^2)/(-k + k*tan(b/2 + k*x/2)^2) + 2*k^2*tan(b/2 + k*x/2)^2*(1 + tan(b/2 + k*x/2)^2)/(-k + k*tan(b/2 + k*x/2)^2)^2, True)), (x, 0, 1)), True))
Use the examples entering the upper and lower limits of integration.