Integración por partes

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Mathematics

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8th Grade

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Easy

Lucia Baez

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8 Slides • 6 Questions

Integración por partes

Open Ended

Si tenemos $\int_{ }^{ }x^2\cos\left(x\right)dx$ a quién sugerimos como u y quién como dv

Type response here

Multiple Choice

Si para $\int_{ }^{ }x^2\cos\left(x\right)dx$ decimos que u= $x^2$ y dv= cos (x)dx, entonces du y v serán...

du=2x dx, v= $\int_{ }^{ }\cos\left(x\right)dx=\sin\left(x\right)$

$du=\frac{x^3}{3}\ dx$ , v= cos(x)

du= 2 dx, v= $\int_{ }^{ }\sin\left(x\right)dx=-\cos\left(x\right)$

$\int_{ }^{ }x^2\cos\left(x\right)dx$

aplicando la fórmula $\int_{ }^{ }u\cdot dv=u\cdot v-\int_{ }^{ }vdu$

$\int_{ }^{ }x^2\cos\left(x\right)dx$

$\int_{ }^{ }u\cdot dv=u\cdot v-\int_{ }^{ }vdu$
= $x^2$ sin(x)- $\int_{ }^{ }\sin\left(x\right)2x\ dx$

tenemos que: $\int_{ }^{ }2x\ \sin\left(x\right)\ dx$

La integral es más sencilla que la inicial, pero no es inmediata, por tanto debemos resolver por partes

Entonces: $\int_{ }^{ }2x\sin\left(x\right)dx$

por la propiedad de la integral $2\int x\sin\left(x\right)\ dx$
para esta integral u=x por lo que du= dx
dv= sin(x)dx por lo que v= $\int_{ }^{ }\sin\left(x\right)\ dx=-\cos\left(x\right)$

lo anterior sustituido en $\int_{ }^{ }udv=uv-\int_{ }^{ }vdu$

= -xcos(x)- $\int_{ }^{ }-\cos\left(x\right)dx$
=-xcos(x)+sin (x)
entonces, sustituimos este último resultado en la integral

$x^2\sin\left(x\right)-\int_{ }^{ }2x\sin\left(x\right)dx=$

$x^2\sin\left(x\right)-2\left(x\cos\left(x\right)+\sin\left(x\right)\right)+c$

Open Ended

Si tenemos $\int_{ }^{ }x^2e^xdx$ escribe u, du, dv, v

Type response here

Open Ended

Si en la integral anterior, u=x²,du=2xdx, dv=e^x, v=e^x, sustituye en la fórmula de integración por partes

Type response here

Open Ended

Notarás que $\int_{ }^{ }e^x2x\ dx$ no es una integral inmediata, por lo que hay que resolver por partes, entonces escribe u, du, dv, v

Type response here

Open Ended

Si definimos que u=2x, du=2dx, dv=e^xdx, v=e^x sustituye en la fórmula de integración por partes

Type response here

Entonces $x^2e^x-\int_{ }^{ }e^x2x\ dx$ =

$x^2e^x-2xe^x-2e^x+c$

Integración por partes

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