Sr Examen
Integral de sqr(1+sin(x)^4) dx
Solución
Ha introducido
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1 / | | 2 | / 4 \ | \1 + sin (x)/ dx | / 0
$$\int\limits_{0}^{1} \left(\sin^{4}{\left(x \right)} + 1\right)^{2}\, dx$$
Integral((1 + sin(x)^4)^2, (x, 0, 1))
Respuesta (Indefinida)
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/ | | 2 3 | / 4 \ 3*sin(2*x) sin (2*x) sin(8*x) 15*sin(4*x) 259*x | \1 + sin (x)/ dx = C - ---------- + --------- + -------- + ----------- + ----- | 4 24 1024 128 128 /
$$\int \left(\sin^{4}{\left(x \right)} + 1\right)^{2}\, dx = C + \frac{259 x}{128} + \frac{\sin^{3}{\left(2 x \right)}}{24} - \frac{3 \sin{\left(2 x \right)}}{4} + \frac{15 \sin{\left(4 x \right)}}{128} + \frac{\sin{\left(8 x \right)}}{1024}$$
Gráfica
Respuesta
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4 4 8 8 3 5 5 3 7 7 3 3 2 2 2 6 6 2 4 4
3*cos (1) 3*sin (1) 35*cos (1) 35*sin (1) 511*cos (1)*sin (1) 385*cos (1)*sin (1) 93*sin (1)*cos(1) 35*cos (1)*sin(1) 5*sin (1)*cos(1) 3*cos (1)*sin(1) 3*cos (1)*sin (1) 35*cos (1)*sin (1) 35*cos (1)*sin (1) 105*cos (1)*sin (1)
1 + --------- + --------- + ---------- + ---------- - ------------------- - ------------------- - ----------------- - ----------------- - ---------------- - ---------------- + ----------------- + ------------------ + ------------------ + -------------------
4 4 128 128 384 384 128 128 4 4 2 32 32 64
$$- \frac{5 \sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{4} - \frac{93 \sin^{7}{\left(1 \right)} \cos{\left(1 \right)}}{128} - \frac{3 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{4} - \frac{511 \sin^{5}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{384} - \frac{385 \sin^{3}{\left(1 \right)} \cos^{5}{\left(1 \right)}}{384} - \frac{35 \sin{\left(1 \right)} \cos^{7}{\left(1 \right)}}{128} + \frac{35 \cos^{8}{\left(1 \right)}}{128} + \frac{35 \sin^{2}{\left(1 \right)} \cos^{6}{\left(1 \right)}}{32} + \frac{3 \cos^{4}{\left(1 \right)}}{4} + \frac{35 \sin^{8}{\left(1 \right)}}{128} + \frac{105 \sin^{4}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{64} + \frac{35 \sin^{6}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{3 \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin^{4}{\left(1 \right)}}{4} + 1$$
=
=
4 4 8 8 3 5 5 3 7 7 3 3 2 2 2 6 6 2 4 4
3*cos (1) 3*sin (1) 35*cos (1) 35*sin (1) 511*cos (1)*sin (1) 385*cos (1)*sin (1) 93*sin (1)*cos(1) 35*cos (1)*sin(1) 5*sin (1)*cos(1) 3*cos (1)*sin(1) 3*cos (1)*sin (1) 35*cos (1)*sin (1) 35*cos (1)*sin (1) 105*cos (1)*sin (1)
1 + --------- + --------- + ---------- + ---------- - ------------------- - ------------------- - ----------------- - ----------------- - ---------------- - ---------------- + ----------------- + ------------------ + ------------------ + -------------------
4 4 128 128 384 384 128 128 4 4 2 32 32 64
$$- \frac{5 \sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{4} - \frac{93 \sin^{7}{\left(1 \right)} \cos{\left(1 \right)}}{128} - \frac{3 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{4} - \frac{511 \sin^{5}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{384} - \frac{385 \sin^{3}{\left(1 \right)} \cos^{5}{\left(1 \right)}}{384} - \frac{35 \sin{\left(1 \right)} \cos^{7}{\left(1 \right)}}{128} + \frac{35 \cos^{8}{\left(1 \right)}}{128} + \frac{35 \sin^{2}{\left(1 \right)} \cos^{6}{\left(1 \right)}}{32} + \frac{3 \cos^{4}{\left(1 \right)}}{4} + \frac{35 \sin^{8}{\left(1 \right)}}{128} + \frac{105 \sin^{4}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{64} + \frac{35 \sin^{6}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{3 \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin^{4}{\left(1 \right)}}{4} + 1$$
1 + 3*cos(1)^4/4 + 3*sin(1)^4/4 + 35*cos(1)^8/128 + 35*sin(1)^8/128 - 511*cos(1)^3*sin(1)^5/384 - 385*cos(1)^5*sin(1)^3/384 - 93*sin(1)^7*cos(1)/128 - 35*cos(1)^7*sin(1)/128 - 5*sin(1)^3*cos(1)/4 - 3*cos(1)^3*sin(1)/4 + 3*cos(1)^2*sin(1)^2/2 + 35*cos(1)^2*sin(1)^6/32 + 35*cos(1)^6*sin(1)^2/32 + 105*cos(1)^4*sin(1)^4/64
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.