Sr Examen
Integral de (cosx-sinx)/(1+sinx)^2 dx
Solución
Ha introducido
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1 / | | cos(x) - sin(x) | --------------- dx | 2 | (1 + sin(x)) | / 0
$$\int\limits_{0}^{1} \frac{- \sin{\left(x \right)} + \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}\, dx$$
Integral((cos(x) - sin(x))/(1 + sin(x))^2, (x, 0, 1))
Respuesta (Indefinida)
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/ /x\ | 6*tan|-| | cos(x) - sin(x) 1 2 \2/ | --------------- dx = C - ---------- + ------------------------------------ + ------------------------------------ | 2 1 + sin(x) 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/ /
$$\int \frac{- \sin{\left(x \right)} + \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}\, dx = C + \frac{6 \tan{\left(\frac{x}{2} \right)}}{3 \tan^{3}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} + 9 \tan{\left(\frac{x}{2} \right)} + 3} + \frac{2}{3 \tan^{3}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} + 9 \tan{\left(\frac{x}{2} \right)} + 3} - \frac{1}{\sin{\left(x \right)} + 1}$$
Gráfica
Respuesta
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2
2 2 6*tan (1/2) 12*tan(1/2)
- - + ------------------------------------------ + ------------------------------------------ + ------------------------------------------
3 3 2 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) 3 + 3*tan (1/2) + 9*tan (1/2) + 9*tan(1/2)
$$- \frac{2}{3} + \frac{6 \tan^{2}{\left(\frac{1}{2} \right)}}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} + \frac{2}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} + \frac{12 \tan{\left(\frac{1}{2} \right)}}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}}$$
=
=
2
2 2 6*tan (1/2) 12*tan(1/2)
- - + ------------------------------------------ + ------------------------------------------ + ------------------------------------------
3 3 2 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) 3 + 3*tan (1/2) + 9*tan (1/2) + 9*tan(1/2)
$$- \frac{2}{3} + \frac{6 \tan^{2}{\left(\frac{1}{2} \right)}}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} + \frac{2}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}} + \frac{12 \tan{\left(\frac{1}{2} \right)}}{3 \tan^{3}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)} + 3 + 9 \tan{\left(\frac{1}{2} \right)}}$$
-2/3 + 2/(3 + 3*tan(1/2)^3 + 9*tan(1/2)^2 + 9*tan(1/2)) + 6*tan(1/2)^2/(3 + 3*tan(1/2)^3 + 9*tan(1/2)^2 + 9*tan(1/2)) + 12*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.