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