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