Questions: Calcular la integral indefinida. [ int fraca x^4-b x^3-c x^2x^5 d x ] A) a ln x+fracbx+fracc2 x^2+C B) a ln x+fracbx-fraccx^2+C C) a ln x-fracbx-fraccx^2+C D) a ln x-fracbx+fracc2 x^2+C

Calcular la integral indefinida. [ int fraca x^4-b x^3-c x^2x^5 d x ] A) a ln x+fracbx+fracc2 x^2+C B) a ln x+fracbx-fraccx^2+C C) a ln x-fracbx-fraccx^2+C D) a ln x-fracbx+fracc2 x^2+C
Transcript text: Calcular la integral indefinida. \[ \int \frac{a x^{4}-b x^{3}-c x^{2}}{x^{5}} d x \] A) $a \ln x+\frac{b}{x}+\frac{c}{2 x^{2}}+C$ B) $a \ln x+\frac{b}{x}-\frac{c}{x^{2}}+C$ C) $a \ln x-\frac{b}{x}-\frac{c}{x^{2}}+C$ D) $a \ln x-\frac{b}{x}+\frac{c}{2 x^{2}}+C$
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Solution

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To solve the integral, we first simplify the integrand by dividing each term in the numerator by x5x^5. This will give us a sum of simpler terms that we can integrate individually. After simplifying, we integrate each term using standard integration rules.

Paso 1: Simplificar el integrando

Primero, simplificamos el integrando dividiendo cada término del numerador por x5x^5: ax4−bx3−cx2x5=ax4x5−bx3x5−cx2x5=ax−bx2−cx3 \frac{a x^{4} - b x^{3} - c x^{2}}{x^{5}} = \frac{a x^{4}}{x^{5}} - \frac{b x^{3}}{x^{5}} - \frac{c x^{2}}{x^{5}} = \frac{a}{x} - \frac{b}{x^{2}} - \frac{c}{x^{3}}

Paso 2: Integrar cada término

Integramos cada término por separado: ∫(ax−bx2−cx3)dx \int \left( \frac{a}{x} - \frac{b}{x^{2}} - \frac{c}{x^{3}} \right) dx

Para el primer término: ∫axdx=aln⁡x \int \frac{a}{x} dx = a \ln x

Para el segundo término: ∫−bx2dx=b(−1x)=bx \int -\frac{b}{x^{2}} dx = b \left( -\frac{1}{x} \right) = \frac{b}{x}

Para el tercer término: ∫−cx3dx=c(−12x2)=−c2x2 \int -\frac{c}{x^{3}} dx = c \left( -\frac{1}{2x^{2}} \right) = -\frac{c}{2x^{2}}

Paso 3: Combinar los resultados

Combinamos los resultados de las integrales: aln⁡x+bx−c2x2+C a \ln x + \frac{b}{x} - \frac{c}{2 x^{2}} + C

Respuesta Final

La respuesta correcta es: D \boxed{D}

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