The problem asks which of the given integrals requires integration by parts only once. The options are: A. $\int (1-x^2)e^{2x} dx$ B. $\int x^2e^{2x} dx$ C. $\int (1-x)e^{2x} dx$ D. $\int -x^2e^{2x} dx$

1. Problem Description

The problem asks which of the given integrals requires integration by parts only once. The options are:

A. $\int (1-x^2)e^{2x} dx$

B. $\int x^2e^{2x} dx$

C. $\int (1-x)e^{2x} dx$

D. $\int -x^2e^{2x} dx$

2. Solution Steps

We need to determine how many times integration by parts is required for each integral. The number of times we need to apply integration by parts is generally determined by the power of $x$ in the polynomial term.

The general formula for integration by parts is:

$\int u dv = uv - \int v du$

A. $\int (1-x^2)e^{2x} dx$. Here, $1-x^2$ is a polynomial of degree

2. Applying integration by parts once will reduce the power to 1, and applying it again will reduce the power to

0. Thus, it requires integration by parts twice.

B. $\int x^2e^{2x} dx$. Here, $x^2$ is a polynomial of degree

2. Applying integration by parts once will reduce the power to 1, and applying it again will reduce the power to

0. Thus, it requires integration by parts twice.

C. $\int (1-x)e^{2x} dx$. Here, $1-x$ is a polynomial of degree

1. Applying integration by parts once will reduce the power to

0. Thus, it requires integration by parts once.

D. $\int -x^2e^{2x} dx$. Here, $-x^2$ is a polynomial of degree

2. Applying integration by parts once will reduce the power to 1, and applying it again will reduce the power to

0. Thus, it requires integration by parts twice.

Therefore, the integral in option C requires integration by parts only once.

3. Final Answer

C. $\int (1-x)e^{2x} dx$