Sr Examen
Integral de (x-arctg2x)/(1-4x^2) dx
Solución
Ha introducido
[src]
1/2 / | | x - atan(2*x) | ------------- dx | 2 | 1 - 4*x | / 0
$$\int\limits_{0}^{\frac{1}{2}} \frac{x - \operatorname{atan}{\left(2 x \right)}}{1 - 4 x^{2}}\, dx$$
Integral((x - atan(2*x))/(1 - 4*x^2), (x, 0, 1/2))
Respuesta (Indefinida)
[src]
// / \
|| | |
|| | acoth(2*x) |
|| - | ---------- dx |
|| | 2 |
|| | 1 + 4*x |
|| | |
|| / |
|| ------------------ for x < -1/2|
|| 2 |
|| |
|| -1/2 -1/2 |
|| / / |
|| | | / |
|| | atanh(2*x) | acoth(2*x) | |
/ || | ---------- dx | ---------- dx | atanh(2*x) | //-acoth(2*x) 2 \
| || | 2 | 2 | ---------- dx | / 2\ ||------------ for x > 1/4|
| x - atan(2*x) || | 1 + 4*x | 1 + 4*x | 2 | log\-1 + 4*x / || 2 |
| ------------- dx = C - 2*|< | | | 1 + 4*x | - -------------- + |< |*atan(2*x)
| 2 || / / | | 8 ||-atanh(2*x) 2 |
| 1 - 4*x || / | ||------------ for x < 1/4|
| || ------------------ - ------------------ - ---------------- for x < 1/2 | \\ 2 /
/ || 2 2 2 |
|| |
|| 1/2 -1/2 -1/2 1/2 |
|| / / / / |
|| | | / | | |
|| | acoth(2*x) | atanh(2*x) | | acoth(2*x) | atanh(2*x) |
|| | ---------- dx | ---------- dx | acoth(2*x) | ---------- dx | ---------- dx |
|| | 2 | 2 | ---------- dx | 2 | 2 |
|| | 1 + 4*x | 1 + 4*x | 2 | 1 + 4*x | 1 + 4*x |
|| | | | 1 + 4*x | | |
||/ / | / / |
|| / |
||----------------- + ------------------ - ---------------- - ------------------ - ----------------- otherwise |
|| 2 2 2 2 2 |
\\ /
$$\int \frac{x - \operatorname{atan}{\left(2 x \right)}}{1 - 4 x^{2}}\, dx = C + \left(\begin{cases} - \frac{\operatorname{acoth}{\left(2 x \right)}}{2} & \text{for}\: x^{2} > \frac{1}{4} \\- \frac{\operatorname{atanh}{\left(2 x \right)}}{2} & \text{for}\: x^{2} < \frac{1}{4} \end{cases}\right) \operatorname{atan}{\left(2 x \right)} - 2 \left(\begin{cases} - \frac{\int \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} & \text{for}\: x < - \frac{1}{2} \\- \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} - \frac{\int \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} + \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} & \text{for}\: x < \frac{1}{2} \\- \frac{\int \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} - \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} + \frac{\int\limits^{\frac{1}{2}} \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} + \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} - \frac{\int\limits^{\frac{1}{2}} \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} & \text{otherwise} \end{cases}\right) - \frac{\log{\left(4 x^{2} - 1 \right)}}{8}$$
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.