Sr Examen
Integral de sin(x+pi/4)^2 dx
Solución
Ha introducido
[src]
pi -- 4 / | | 2/ pi\ | sin |x + --| dx | \ 4 / | / 9
$$\int\limits_{9}^{\frac{\pi}{4}} \sin^{2}{\left(x + \frac{\pi}{4} \right)}\, dx$$
Integral(sin(x + pi/4)^2, (x, 9, pi/4))
Respuesta (Indefinida)
[src]
/ /x pi\ 3/x pi\ 4/x pi\ 2/x pi\ | 2*tan|- + --| 2*tan |- + --| x*tan |- + --| 2*x*tan |- + --| | 2/ pi\ x \2 8 / \2 8 / \2 8 / \2 8 / | sin |x + --| dx = C + ----------------------------------- - ----------------------------------- + ----------------------------------- + ----------------------------------- + ----------------------------------- | \ 4 / 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ | 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 8 / \2 8 / \2 8 / \2 8 / \2 8 / \2 8 / \2 8 / \2 8 / \2 8 / \2 8 /
$$\int \sin^{2}{\left(x + \frac{\pi}{4} \right)}\, dx = C + \frac{x \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 2} + \frac{2 \tan^{3}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 2}$$
Gráfica
Respuesta
[src]
2/ pi\ 2/ pi\ / pi\ / pi\
9*cos |9 + --| 9*sin |9 + --| cos|9 + --|*sin|9 + --|
\ 4 / \ 4 / pi \ 4 / \ 4 /
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2 2 8 2
$$- \frac{9 \cos^{2}{\left(\frac{\pi}{4} + 9 \right)}}{2} - \frac{9 \sin^{2}{\left(\frac{\pi}{4} + 9 \right)}}{2} + \frac{\sin{\left(\frac{\pi}{4} + 9 \right)} \cos{\left(\frac{\pi}{4} + 9 \right)}}{2} + \frac{\pi}{8}$$
=
=
2/ pi\ 2/ pi\ / pi\ / pi\
9*cos |9 + --| 9*sin |9 + --| cos|9 + --|*sin|9 + --|
\ 4 / \ 4 / pi \ 4 / \ 4 /
- -------------- - -------------- + -- + -----------------------
2 2 8 2
$$- \frac{9 \cos^{2}{\left(\frac{\pi}{4} + 9 \right)}}{2} - \frac{9 \sin^{2}{\left(\frac{\pi}{4} + 9 \right)}}{2} + \frac{\sin{\left(\frac{\pi}{4} + 9 \right)} \cos{\left(\frac{\pi}{4} + 9 \right)}}{2} + \frac{\pi}{8}$$
-9*cos(9 + pi/4)^2/2 - 9*sin(9 + pi/4)^2/2 + pi/8 + cos(9 + pi/4)*sin(9 + pi/4)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.