Evaluate the indefinite integral:

Question

Evaluate the indefinite integral:

∫ from 1 to 2 of ln (x²e^x) dx

Thank you for your help!!

Answer 1

Since powers of decrease when you take the derivative, let

u = x² so du = 2xdx
dv = e^xx dx so v = e^xx

∫ u dv = uv = ∫ v du
= x²e^x - ∫2xe^x dx

Now let u = x so du = dx
dv = e^xdx so v = e^x

∫2xe^x dx = xe^x - ∫e^x dx = xe^x - e^x

So, ∫x²e^x = x²e^x - 2(xe^x - e^x)
= e^x(x² - 2x + 2)

Answer 2

I see there were some typos. In correct form,

u = x² so du = 2xdx
dv = e^x dx so v = e^x

∫ u dv = uv = ∫ v du
= x²e^x - ∫2xe^x dx

Now let u = x so du = dx
dv = e^xdx so v = e^x

∫xe^x dx = xe^x - ∫e^x dx = xe^x - e^x

So, ∫x²e^x = x²e^x - 2(xe^x - e^x)
= e^x(x² - 2x + 2)

Answer 3

To evaluate the given indefinite integral ∫ (1 to 2) ln(x²e^x) dx, we need to use integration techniques. In this case, we can use the properties of logarithms and integration by parts.

Let's start by simplifying the integrand. Using the properties of logarithms, we can rewrite ln(x²e^x) as ln(x²) + ln(e^x).

Now, we have ∫ (1 to 2) ln(x²) + ln(e^x) dx.

The integral of ln(x²) can be found by using the substitution method. Let's substitute u = x², then du = 2x dx. Rearranging this equation, we get dx = du / (2x), and substituting these values back into the integral gives us:

∫ (1 to 2) ln(u) du / (2x).

Since we are integrating with respect to u, we need to express x in terms of u. Taking the square root of u, we have x = √u. Thus, the integral becomes:

∫ (1 to 2) ln(u) du / (2√u).

Next, let's focus on the integral of ln(e^x). This simplifies to just x, so we have:

∫ (1 to 2) x dx.

Now, we can integrate each term separately.

For the first term, ∫ ln(u) du / (2√u), we can simplify further by using the property of logarithms ln(u) = ln(u^(1/2)), which can be replaced with (1/2)ln(u). Therefore, we have:

(1/2) ∫ (1 to 2) ln(u) du / (2√u).

Now, let's integrate the second term, ∫ x dx, which gives us:

(1/2) [x²/2] (1 to 2).

Simplifying both terms, we have:

(1/4) ∫ (1 to 2) ln(u) du / √u + (1/4) [x²/2] (1 to 2).

Evaluating the definite integral, we get:

(1/4) [(2²/2)ln(2) / √2 - (1²/2)ln(1) / √1] + (1/4) [(2²/2) - (1²/2)].

Calculating further,

= (1/4) [2 ln(2) / √2 - 0] + (1/4) [2 - 1]
= (1/4) (2 ln(2) / √2 + 1).

So, the value of the indefinite integral ∫ (1 to 2) ln(x²e^x) dx is (1/4) (2 ln(2) / √2 + 1).

Please double-check the calculations, as there might be possible errors.