Question

Use one of the reduction formulas shown to the right (which are valid for $a \neq 0$) to evaluate the integral below. $\int x^5 e^{5x} dx$ $\int x^5 e^{5x} dx = \square$ $\int x^n e^{ax} dx = \frac{x^n e^{ax}}{a} - \frac{n}{a} \int x^{n-1} e^{ax} dx$ $\int x^n \cos ax \, dx = \frac{x^n \sin ax}{a} - \frac{n}{a} \int x^{n-1} \sin ax \, dx$ $\int x^n \sin ax \, dx = -\frac{x^n \cos ax}{a} + \frac{n}{a} \int x^{n-1} \cos ax \, dx$ $\int \ln^n x \, dx = x \ln^n x - n \int \ln^{n-1} x \, dx$

          Use one of the reduction formulas shown to the right (which are valid for $a \neq 0$) to evaluate the integral below.
$\int x^5 e^{5x} dx$
$\int x^5 e^{5x} dx = \square$
$\int x^n e^{ax} dx = \frac{x^n e^{ax}}{a} - \frac{n}{a} \int x^{n-1} e^{ax} dx$
$\int x^n \cos ax \, dx = \frac{x^n \sin ax}{a} - \frac{n}{a} \int x^{n-1} \sin ax \, dx$
$\int x^n \sin ax \, dx = -\frac{x^n \cos ax}{a} + \frac{n}{a} \int x^{n-1} \cos ax \, dx$
$\int \ln^n x \, dx = x \ln^n x - n \int \ln^{n-1} x \, dx$

Use one of the reduction formulas shown to the right (which are valid for a ≠ 0) to evaluate the integral below.
∫ x^5 e^5x dx
∫ x^5 e^5x dx = □
∫ x^n e^ax dx = (x^n e^ax)/(a) - (n)/(a)∫ x^n-1 e^ax dx
∫ x^n cos ax dx = (x^n sin ax)/(a) - (n)/(a)∫ x^n-1sin ax dx
∫ x^n sin ax dx = -(x^n cos ax)/(a) + (n)/(a)∫ x^n-1cos ax dx
∫ln^n x dx = x ln^n x - n ∫ln^n-1 x dx

Added by Amanda C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/31/2023

Step 1

In this case, n = 5 and a = 5. Show more…

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Use one of the reduction formulas shown to the right (which are valid for a ≠ 0) to evaluate the integral below: ∫ x^(n)e^(ax)dx = (x^(n)e^(ax))/(a) - (n)/(a)∫ x^(n-1)e^(ax)dx ∫ x^(n)cos(ax)dx = (x^(n)sin(ax))/(a) - (n)/(a)∫ x^(n-1)sin(ax)dx ∫ x^(n)sin(ax)dx = -(x^(n)cos(ax))/(a) + (n)/(a)∫ x^(n-1)cos(ax)dx ∫ ln^(n)(x)dx = xln^(n)(x) - ∫^0 ln^(n-1)(x)dx ∫ x^(5)e^(5x)dx = Use one of the reduction formulas shown to the right (which are valid for a ≠ 0) to evaluate the integral below: ∫ x^(n)xdx = xln(x) - n∫x^(n-1)dx