Sr Examen
Integral de sqrt(a-bx^2) dx
Solución
Ha introducido
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1 / | | __________ | / 2 | \/ a - b*x dx | / 0
$$\int\limits_{0}^{1} \sqrt{a - b x^{2}}\, dx$$
Integral(sqrt(a - b*x^2), (x, 0, 1))
Respuesta (Indefinida)
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// / ___\ \
|| |x*\/ b | |
|| I*a*acosh|-------| |
|| | ___ | ___ 3 | 2| |
|| \ \/ a / I*x*\/ a I*b*x |b*x | |
||- ------------------ - ------------------ + ------------------------ for |----| > 1|
/ || ___ ___________ ___________ | a | |
| || 2*\/ b / 2 / 2 |
| __________ || / b*x ___ / b*x |
| / 2 || 2* / -1 + ---- 2*\/ a * / -1 + ---- |
| \/ a - b*x dx = C + |< \/ a \/ a |
| || |
/ || / ___\ __________ |
|| |x*\/ b | / 2 |
|| a*asin|-------| ___ / b*x |
|| | ___ | x*\/ a * / 1 - ---- |
|| \ \/ a / \/ a |
|| --------------- + ----------------------- otherwise |
|| ___ 2 |
|| 2*\/ b |
\\ /
$$\int \sqrt{a - b x^{2}}\, dx = C + \begin{cases} - \frac{i \sqrt{a} x}{2 \sqrt{-1 + \frac{b x^{2}}{a}}} - \frac{i a \operatorname{acosh}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} + \frac{i b x^{3}}{2 \sqrt{a} \sqrt{-1 + \frac{b x^{2}}{a}}} & \text{for}\: \left|{\frac{b x^{2}}{a}}\right| > 1 \\\frac{\sqrt{a} x \sqrt{1 - \frac{b x^{2}}{a}}}{2} + \frac{a \operatorname{asin}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} & \text{otherwise} \end{cases}$$
Respuesta
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1 / | | / ___ 2 2 4 2 | | I*\/ a I*b*x I*b *x 3*I*b*x 2 |b| | |- ---------------- + ---------------------- - --------------------- + ------------------------ for x *|-| > 1 | | ___________ 3/2 3/2 ___________ |a| | | / 2 / 2\ / 2\ / 2 | | / b*x ___ | b*x | 3/2 | b*x | ___ / b*x | | / -1 + ---- 2*\/ a *|-1 + ----| 2*a *|-1 + ----| 2*\/ a * / -1 + ---- | | \/ a \ a / \ a / \/ a | | | | __________ | < / 2 dx | | ___ / b*x | | \/ a * / 1 - ---- ___ 2 | | \/ a \/ a b*x | | --------------------- + ----------------- - ----------------------- otherwise | | 2 __________ __________ | | / 2 / 2 | | / b*x ___ / b*x | | 2* / 1 - ---- 2*\/ a * / 1 - ---- | | \/ a \/ a | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i \sqrt{a}}{\sqrt{-1 + \frac{b x^{2}}{a}}} + \frac{3 i b x^{2}}{2 \sqrt{a} \sqrt{-1 + \frac{b x^{2}}{a}}} + \frac{i b x^{2}}{2 \sqrt{a} \left(-1 + \frac{b x^{2}}{a}\right)^{\frac{3}{2}}} - \frac{i b^{2} x^{4}}{2 a^{\frac{3}{2}} \left(-1 + \frac{b x^{2}}{a}\right)^{\frac{3}{2}}} & \text{for}\: x^{2} \left|{\frac{b}{a}}\right| > 1 \\\frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}}}{2} + \frac{\sqrt{a}}{2 \sqrt{1 - \frac{b x^{2}}{a}}} - \frac{b x^{2}}{2 \sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / ___ 2 2 4 2 | | I*\/ a I*b*x I*b *x 3*I*b*x 2 |b| | |- ---------------- + ---------------------- - --------------------- + ------------------------ for x *|-| > 1 | | ___________ 3/2 3/2 ___________ |a| | | / 2 / 2\ / 2\ / 2 | | / b*x ___ | b*x | 3/2 | b*x | ___ / b*x | | / -1 + ---- 2*\/ a *|-1 + ----| 2*a *|-1 + ----| 2*\/ a * / -1 + ---- | | \/ a \ a / \ a / \/ a | | | | __________ | < / 2 dx | | ___ / b*x | | \/ a * / 1 - ---- ___ 2 | | \/ a \/ a b*x | | --------------------- + ----------------- - ----------------------- otherwise | | 2 __________ __________ | | / 2 / 2 | | / b*x ___ / b*x | | 2* / 1 - ---- 2*\/ a * / 1 - ---- | | \/ a \/ a | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i \sqrt{a}}{\sqrt{-1 + \frac{b x^{2}}{a}}} + \frac{3 i b x^{2}}{2 \sqrt{a} \sqrt{-1 + \frac{b x^{2}}{a}}} + \frac{i b x^{2}}{2 \sqrt{a} \left(-1 + \frac{b x^{2}}{a}\right)^{\frac{3}{2}}} - \frac{i b^{2} x^{4}}{2 a^{\frac{3}{2}} \left(-1 + \frac{b x^{2}}{a}\right)^{\frac{3}{2}}} & \text{for}\: x^{2} \left|{\frac{b}{a}}\right| > 1 \\\frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}}}{2} + \frac{\sqrt{a}}{2 \sqrt{1 - \frac{b x^{2}}{a}}} - \frac{b x^{2}}{2 \sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i*sqrt(a)/sqrt(-1 + b*x^2/a) + i*b*x^2/(2*sqrt(a)*(-1 + b*x^2/a)^(3/2)) - i*b^2*x^4/(2*a^(3/2)*(-1 + b*x^2/a)^(3/2)) + 3*i*b*x^2/(2*sqrt(a)*sqrt(-1 + b*x^2/a)), x^2*|b/a| > 1), (sqrt(a)*sqrt(1 - b*x^2/a)/2 + sqrt(a)/(2*sqrt(1 - b*x^2/a)) - b*x^2/(2*sqrt(a)*sqrt(1 - b*x^2/a)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.