Sr Examen
Integral de dx/(a+bsinx) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------ dx | a + b*sin(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{a + b \sin{\left(x \right)}}\, dx$$
Integral(1/(a + b*sin(x)), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / /x\\ \
|| zoo*log|tan|-|| for And(a = 0, b = 0)|
|| \ \2// |
|| |
|| 2 |
|| ------------- for a = -b |
|| /x\ |
|| -b + b*tan|-| |
|| \2/ |
|| |
|| -2 |
|| ------------ for a = b |
/ || /x\ |
| || b + b*tan|-| |
| 1 || \2/ |
| ------------ dx = C + |< |
| a + b*sin(x) || / /x\\ |
| || log|tan|-|| |
/ || \ \2// |
|| ----------- for a = 0 |
|| b |
|| |
|| / _________ \ / _________ \ |
|| _________ | / 2 2 | _________ | / 2 2 | |
|| / 2 2 |b \/ b - a /x\| / 2 2 |b \/ b - a /x\| |
||\/ b - a *log|- + ------------ + tan|-|| \/ b - a *log|- - ------------ + tan|-|| |
|| \a a \2// \a a \2// |
||------------------------------------------- - ------------------------------------------- otherwise |
|| 2 2 2 2 |
|| a - b a - b |
\\ /
$$\int \frac{1}{a + b \sin{\left(x \right)}}\, dx = C + \begin{cases} \tilde{\infty} \log{\left(\tan{\left(\frac{x}{2} \right)} \right)} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2}{b \tan{\left(\frac{x}{2} \right)} - b} & \text{for}\: a = - b \\- \frac{2}{b \tan{\left(\frac{x}{2} \right)} + b} & \text{for}\: a = b \\\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{b} & \text{for}\: a = 0 \\- \frac{\sqrt{- a^{2} + b^{2}} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{a^{2} - b^{2}} + \frac{\sqrt{- a^{2} + b^{2}} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ ____ | / 2 / / ____\ / ____ ____\ / ____ ____\ ____\ | 2 2*\/ b | | / 2 | | / 2 / 2 | | / 2 / 2 | / 2 | | - + ----------------------- for Or\And\a = 0, a = -\/ b /, And\a = -\/ b , a = \/ b /, And\a = 0, a = -\/ b , a = \/ b /, a = -\/ b / | b ____ | 2 / 2 | b *tan(1/2) - b*\/ b | | ____ | / 2 / / ____\ ____\ | 2 2*\/ b | | / 2 | / 2 | | - - ----------------------- for Or\And\a = 0, a = \/ b /, a = \/ b / | b ____ | / 2 2 < b*\/ b + b *tan(1/2) | | /1\ log(tan(1/2)) | oo*sign|-| + ------------- for a = 0 | \b/ b | | / _________\ / _________ \ / _________\ / _________ \ | | / 2 2 | | / 2 2 | | / 2 2 | | / 2 2 | | |b \/ b - a | |b \/ b - a | |b \/ b - a | |b \/ b - a | |log|- + ------------| log|- - ------------ + tan(1/2)| log|- - ------------| log|- + ------------ + tan(1/2)| | \a a / \a a / \a a / \a a / |--------------------- + -------------------------------- - --------------------- - -------------------------------- otherwise | _________ _________ _________ _________ | / 2 2 / 2 2 / 2 2 / 2 2 \ \/ b - a \/ b - a \/ b - a \/ b - a
$$\begin{cases} \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} - b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - \sqrt{b^{2}}\right) \vee \left(a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee \left(a = 0 \wedge a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee a = - \sqrt{b^{2}} \\- \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} + b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = \sqrt{b^{2}}\right) \vee a = \sqrt{b^{2}} \\\infty \operatorname{sign}{\left(\frac{1}{b} \right)} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} \right)}}{b} & \text{for}\: a = 0 \\- \frac{\log{\left(\frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} & \text{otherwise} \end{cases}$$
=
=
/ ____ | / 2 / / ____\ / ____ ____\ / ____ ____\ ____\ | 2 2*\/ b | | / 2 | | / 2 / 2 | | / 2 / 2 | / 2 | | - + ----------------------- for Or\And\a = 0, a = -\/ b /, And\a = -\/ b , a = \/ b /, And\a = 0, a = -\/ b , a = \/ b /, a = -\/ b / | b ____ | 2 / 2 | b *tan(1/2) - b*\/ b | | ____ | / 2 / / ____\ ____\ | 2 2*\/ b | | / 2 | / 2 | | - - ----------------------- for Or\And\a = 0, a = \/ b /, a = \/ b / | b ____ | / 2 2 < b*\/ b + b *tan(1/2) | | /1\ log(tan(1/2)) | oo*sign|-| + ------------- for a = 0 | \b/ b | | / _________\ / _________ \ / _________\ / _________ \ | | / 2 2 | | / 2 2 | | / 2 2 | | / 2 2 | | |b \/ b - a | |b \/ b - a | |b \/ b - a | |b \/ b - a | |log|- + ------------| log|- - ------------ + tan(1/2)| log|- - ------------| log|- + ------------ + tan(1/2)| | \a a / \a a / \a a / \a a / |--------------------- + -------------------------------- - --------------------- - -------------------------------- otherwise | _________ _________ _________ _________ | / 2 2 / 2 2 / 2 2 / 2 2 \ \/ b - a \/ b - a \/ b - a \/ b - a
$$\begin{cases} \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} - b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - \sqrt{b^{2}}\right) \vee \left(a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee \left(a = 0 \wedge a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee a = - \sqrt{b^{2}} \\- \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} + b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = \sqrt{b^{2}}\right) \vee a = \sqrt{b^{2}} \\\infty \operatorname{sign}{\left(\frac{1}{b} \right)} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} \right)}}{b} & \text{for}\: a = 0 \\- \frac{\log{\left(\frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} & \text{otherwise} \end{cases}$$
Piecewise((2/b + 2*sqrt(b^2)/(b^2*tan(1/2) - b*sqrt(b^2)), (a = -sqrt(b^2))∨((a = 0)∧(a = -sqrt(b^2)))∨((a = sqrt(b^2))∧(a = -sqrt(b^2)))∨((a = 0)∧(a = sqrt(b^2))∧(a = -sqrt(b^2)))), (2/b - 2*sqrt(b^2)/(b*sqrt(b^2) + b^2*tan(1/2)), (a = sqrt(b^2))∨((a = 0)∧(a = sqrt(b^2)))), (oo*sign(1/b) + log(tan(1/2))/b, a = 0), (log(b/a + sqrt(b^2 - a^2)/a)/sqrt(b^2 - a^2) + log(b/a - sqrt(b^2 - a^2)/a + tan(1/2))/sqrt(b^2 - a^2) - log(b/a - sqrt(b^2 - a^2)/a)/sqrt(b^2 - a^2) - log(b/a + sqrt(b^2 - a^2)/a + tan(1/2))/sqrt(b^2 - a^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.