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