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- Integral of d{x}:
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Integral of 1/(x*(1-x))
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Integral of x/(x^3+8)
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Integral of x^(-6)
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Integral of x^3e^x
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Identical expressions
- (dx)/(cos^ two (ax+b))
- (dx) divide by ( co sinus of e of squared (ax plus b))
- (dx) divide by ( co sinus of e of to the power of two (ax plus b))
- (dx)/(cos2(ax+b))
- dx/cos2ax+b
- (dx)/(cos²(ax+b))
- (dx)/(cos to the power of 2(ax+b))
- dx/cos^2ax+b
- (dx) divide by (cos^2(ax+b))
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Similar expressions
- (dx)/(cos^2(ax-b))
Integral of (dx)/(cos^2(ax+b)) dx
The solution
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[src]
1 / | | 1 | ------------- dx | 2 | cos (a*x + b) | / 0
$$\int\limits_{0}^{1} \frac{1}{\cos^{2}{\left(a x + b \right)}}\, dx$$
Integral(1/(cos(a*x + b)^2), (x, 0, 1))
The answer (Indefinite)
[src]
// / -pi \\
|| zoo*x for And|a = 0, b = ----||
|| \ 2 /|
|| |
|| x |
|| ------- for a = 0 |
|| 2 |
|| cos (b) |
/ || |
| || / pi\ |
| 1 || -|b + --| |
| ------------- dx = C + |< \ 2 / |
| 2 || zoo*x for a = ---------- |
| cos (a*x + b) || x |
| || |
/ || /b a*x\ |
|| -2*tan|- + ---| |
|| \2 2 / |
||-------------------- otherwise |
|| 2/b a*x\ |
||-a + a*tan |- + ---| |
|| \2 2 / |
\\ /
$$\int \frac{1}{\cos^{2}{\left(a x + b \right)}}\, dx = C + \begin{cases} \tilde{\infty} x & \text{for}\: a = 0 \wedge b = - \frac{\pi}{2} \\\frac{x}{\cos^{2}{\left(b \right)}} & \text{for}\: a = 0 \\\tilde{\infty} x & \text{for}\: a = - \frac{b + \frac{\pi}{2}}{x} \\- \frac{2 \tan{\left(\frac{a x}{2} + \frac{b}{2} \right)}}{a \tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} - a} & \text{otherwise} \end{cases}$$
The answer
[src]
/ / -pi \ | nan for And|a = 0, b = ----| | \ 2 / | | 1 | ------- for a = 0 | 2 | cos (b) | | 1 | / | | | | / / pi\ | | | -|b + --| | | | \ 2 / < | | 0 for a = ---------- | | | x | | | | | | / 2/b a*x\\ 2 2/b a*x\ / 2/b a*x\\ | | < a*|1 + tan |- + ---|| 2*a *tan |- + ---|*|1 + tan |- + ---|| dx otherwise | | | \ \2 2 // \2 2 / \ \2 2 // | | |- --------------------- + -------------------------------------- otherwise | | | 2/b a*x\ 2 | | | -a + a*tan |- + ---| / 2/b a*x\\ | | | \2 2 / |-a + a*tan |- + ---|| | | | \ \2 2 // | | \ | | |/ |0 \
$$\begin{cases} \text{NaN} & \text{for}\: a = 0 \wedge b = - \frac{\pi}{2} \\\frac{1}{\cos^{2}{\left(b \right)}} & \text{for}\: a = 0 \\\int\limits_{0}^{1} \begin{cases} 0 & \text{for}\: a = - \frac{b + \frac{\pi}{2}}{x} \\\frac{2 a^{2} \left(\tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} + 1\right) \tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)}}{\left(a \tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} - a\right)^{2}} - \frac{a \left(\tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} + 1\right)}{a \tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} - a} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
=
=
/ / -pi \ | nan for And|a = 0, b = ----| | \ 2 / | | 1 | ------- for a = 0 | 2 | cos (b) | | 1 | / | | | | / / pi\ | | | -|b + --| | | | \ 2 / < | | 0 for a = ---------- | | | x | | | | | | / 2/b a*x\\ 2 2/b a*x\ / 2/b a*x\\ | | < a*|1 + tan |- + ---|| 2*a *tan |- + ---|*|1 + tan |- + ---|| dx otherwise | | | \ \2 2 // \2 2 / \ \2 2 // | | |- --------------------- + -------------------------------------- otherwise | | | 2/b a*x\ 2 | | | -a + a*tan |- + ---| / 2/b a*x\\ | | | \2 2 / |-a + a*tan |- + ---|| | | | \ \2 2 // | | \ | | |/ |0 \
$$\begin{cases} \text{NaN} & \text{for}\: a = 0 \wedge b = - \frac{\pi}{2} \\\frac{1}{\cos^{2}{\left(b \right)}} & \text{for}\: a = 0 \\\int\limits_{0}^{1} \begin{cases} 0 & \text{for}\: a = - \frac{b + \frac{\pi}{2}}{x} \\\frac{2 a^{2} \left(\tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} + 1\right) \tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)}}{\left(a \tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} - a\right)^{2}} - \frac{a \left(\tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} + 1\right)}{a \tan^{2}{\left(\frac{a x}{2} + \frac{b}{2} \right)} - a} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((nan, (a = 0)∧(b = -pi/2)), (cos(b)^(-2), a = 0), (Integral(Piecewise((0, a = -(b + pi/2)/x), (-a*(1 + tan(b/2 + a*x/2)^2)/(-a + a*tan(b/2 + a*x/2)^2) + 2*a^2*tan(b/2 + a*x/2)^2*(1 + tan(b/2 + a*x/2)^2)/(-a + a*tan(b/2 + a*x/2)^2)^2, True)), (x, 0, 1)), True))
Use the examples entering the upper and lower limits of integration.