Sr Examen
Integral de (sinx)/((1+sinx)^2) dx
Solución
Ha introducido
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1 / | | sin(x) | ------------- dx | 2 | (1 + sin(x)) | / 0
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}\, dx$$
Integral(sin(x)/(1 + sin(x))^2, (x, 0, 1))
Respuesta (Indefinida)
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/ /x\ | 6*tan|-| | sin(x) 2 \2/ | ------------- dx = C - ------------------------------------ - ------------------------------------ | 2 3/x\ 2/x\ /x\ 3/x\ 2/x\ /x\ | (1 + sin(x)) 3 + 3*tan |-| + 9*tan |-| + 9*tan|-| 3 + 3*tan |-| + 9*tan |-| + 9*tan|-| | \2/ \2/ \2/ \2/ \2/ \2/ /
$$\int \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}\, dx = C - \frac{6 \tan{\left(\frac{x}{2} \right)}}{3 \tan^{3}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} + 9 \tan{\left(\frac{x}{2} \right)} + 3} - \frac{2}{3 \tan^{3}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} + 9 \tan{\left(\frac{x}{2} \right)} + 3}$$
Gráfica
Respuesta
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2 2 6*tan(1/2)
- - ------------------------------------------ - ------------------------------------------
3 3 2 3 2
3 + 3*tan (1/2) + 9*tan (1/2) + 9*tan(1/2) 3 + 3*tan (1/2) + 9*tan (1/2) + 9*tan(1/2)
$$- \frac{6 \tan{\left(\frac{1}{2} \right)}}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} - \frac{2}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} + \frac{2}{3}$$
=
=
2 2 6*tan(1/2)
- - ------------------------------------------ - ------------------------------------------
3 3 2 3 2
3 + 3*tan (1/2) + 9*tan (1/2) + 9*tan(1/2) 3 + 3*tan (1/2) + 9*tan (1/2) + 9*tan(1/2)
$$- \frac{6 \tan{\left(\frac{1}{2} \right)}}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} - \frac{2}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} + \frac{2}{3}$$
2/3 - 2/(3 + 3*tan(1/2)^3 + 9*tan(1/2)^2 + 9*tan(1/2)) - 6*tan(1/2)/(3 + 3*tan(1/2)^3 + 9*tan(1/2)^2 + 9*tan(1/2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.