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