Sr Examen
Integral de (cos^(3)x)/(1+sinx) dx
Solución
Ha introducido
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1 / | | 3 | cos (x) | ---------- dx | 1 + sin(x) | / 0
$$\int\limits_{0}^{1} \frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)} + 1}\, dx$$
Integral(cos(x)^3/(1 + sin(x)), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ | 2/x\ 3/x\ /x\ | 3 2*tan |-| 2*tan |-| 2*tan|-| | cos (x) \2/ \2/ \2/ | ---------- dx = C - ----------------------- + ----------------------- + ----------------------- | 1 + sin(x) 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ | 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| / \2/ \2/ \2/ \2/ \2/ \2/
$$\int \frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)} + 1}\, dx = C + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
Gráfica
Respuesta
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2 3
2*tan (1/2) 2*tan (1/2) 2*tan(1/2)
- --------------------------- + --------------------------- + ---------------------------
4 2 4 2 4 2
1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2)
$$- \frac{2 \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tan^{3}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1}$$
=
=
2 3
2*tan (1/2) 2*tan (1/2) 2*tan(1/2)
- --------------------------- + --------------------------- + ---------------------------
4 2 4 2 4 2
1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2)
$$- \frac{2 \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tan^{3}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1}$$
-2*tan(1/2)^2/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 2*tan(1/2)^3/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 2*tan(1/2)/(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.