Sr Examen
Integral de sin^4(x/2)cos^2(x/2) dx
Solución
Ha introducido
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1 / | | 4/x\ 2/x\ | sin |-|*cos |-| dx | \2/ \2/ | / 0
$$\int\limits_{0}^{1} \sin^{4}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} \right)}\, dx$$
Integral(sin(x/2)^4*cos(x/2)^2, (x, 0, 1))
Respuesta (Indefinida)
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/ /x\ /x\ 3/x\ /x\ 5/x\ /x\ | cos|-|*sin|-| sin |-|*cos|-| sin |-|*cos|-| | 4/x\ 2/x\ x \2/ \2/ \2/ \2/ \2/ \2/ | sin |-|*cos |-| dx = C + -- - ------------- - -------------- + -------------- | \2/ \2/ 16 8 12 3 | /
$$\int \sin^{4}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} \right)}\, dx = C + \frac{x}{16} + \frac{\sin^{5}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{3} - \frac{\sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{12} - \frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{8}$$
Gráfica
Respuesta
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3 5 1 cos(1/2)*sin(1/2) sin (1/2)*cos(1/2) sin (1/2)*cos(1/2) -- - ----------------- - ------------------ + ------------------ 16 8 12 3
$$- \frac{\sin{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)}}{8} - \frac{\sin^{3}{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)}}{12} + \frac{\sin^{5}{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)}}{3} + \frac{1}{16}$$
=
=
3 5 1 cos(1/2)*sin(1/2) sin (1/2)*cos(1/2) sin (1/2)*cos(1/2) -- - ----------------- - ------------------ + ------------------ 16 8 12 3
$$- \frac{\sin{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)}}{8} - \frac{\sin^{3}{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)}}{12} + \frac{\sin^{5}{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)}}{3} + \frac{1}{16}$$
1/16 - cos(1/2)*sin(1/2)/8 - sin(1/2)^3*cos(1/2)/12 + sin(1/2)^5*cos(1/2)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.