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