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