5. Given : \( \int 6 x^{3} e^{x^{2}+3} d x \). Using integration by parts \( \left(\int u d v=u v-\int v d u\right) \) and letting \( u=f(x) \), \( d v=3 g(x) d x \), find \( h(x) \) if \( \int v d u=h(x)+C \)
a.) \( 6 e^{x^{2}+3} \)
b.) \( 3 e^{-3} e^{x^{2}} \)
c.) \( \frac{3 e^{x^{2}+6}}{e^{3}} \)
d.) \( 2 e^{x^{2}+3} \)
Solve the above problem.

To solve the given integral \( \int 6 x^3 e^{x^2 + 3} \, dx \) using integration by parts, we will first identify the functions \( u \) and \( dv \).

Given: \( \int 6 x^3 e^{x^2 + 3} \, dx \)

We let \( u = 6 x^2 \) and \( dv = x e^{x^2 + 3} \, dx \).

Next, we need to find \( du \) and \( v \):

\[ du = \frac{d}{dx}(6 x^2) \, dx = 12 x \, dx \]
\[ dv = x e^{x^2 + 3} \, dx \]

To find \( v \), we integrate \( dv \):

\[ v = \int x e^{x^2 + 3} \, dx \]

Let \( w = x^2 + 3 \), so \( dw = 2x \, dx \) and \( \frac{dw}{2} = x \, dx \).

Thus,

\[ v = \int x e^{x^2 + 3} \, dx = \int e^w \frac{dw}{2} = \frac{1}{2} \int e^w \, dw = \frac{1}{2} e^w = \frac{1}{2} e^{x^2 + 3} \]

So, now we have:

\[ u = 6 x^2 \]
\[ dv = x e^{x^2 + 3} \, dx \]
\[ du = 12 x \, dx \]
\[ v = \frac{1}{2} e^{x^2 + 3} \]

Using integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get:

\[ \int 6 x^3 e^{x^2 + 3} \, dx = 6 x^2 \cdot \frac{1}{2} e^{x^2 + 3} - \int \left( \frac{1}{2} e^{x^2 + 3} \right) \cdot (12 x) \, dx \]

Simplifying:

\[ = 3 x^2 e^{x^2 + 3} - 6 \int x e^{x^2 + 3} \, dx \]

We already know \( \int x e^{x^2 + 3} \, dx = \frac{1}{2} e^{x^2 + 3} \), so:

\[ = 3 x^2 e^{x^2 + 3} - 6 \cdot \frac{1}{2} e^{x^2 + 3} \]
\[ = 3 x^2 e^{x^2 + 3} - 3 e^{x^2 + 3} \]
\[ = e^{x^2 + 3} (3 x^2 - 3) \]

The integral simplifies to:

\[ \int 6 x^3 e^{x^2 + 3} \, dx = e^{x^2 + 3} (3 x^2 - 3) + C \]

Now, we need to compare this with the options given and identify \( h(x) \). Here, \( h(x) \) represents the term we found within the integral \( e^{x^2 + 3} \).

Comparing with the options, we find:

a.) \( 6 e^{x^2 + 3} \)
b.) \( 3 e^{-3} e^{x^2} \)
c.) \( \frac{3 e^{x^2 + 6}}{e^{3}} \)
d.) \( 2 e^{x^2 + 3} \)

The correct answer is:

\[ \boxed{6 e^{x^2 + 3}} \]

Thus, \( h(x) = 6 e^{x^2 + 3} \).

Learn how to solve the integral of 6x^3 e^(x^2 + 3) using integration by parts with a detailed step-by-step solution. This problem involves advanced calculus techniques and exponential functions. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation