In Using The Technique Of Integration By Parts, You Must Carefully Choose Which Expression Is $u$. Choose $u$. Do Not Evaluate The Integral.$\int X^3 E^{2x} \, Dx$u = \square$

Introduction

Integration by parts is a powerful technique used to solve certain types of integrals. It involves differentiating one function and integrating the other, and then switching the order of differentiation and integration. However, the key to successfully applying integration by parts is to carefully choose which expression is $u$. In this article, we will discuss how to choose the right expression for $u$ in the given integral $\int x^3 e^{2x} \, dx$.

Understanding Integration by Parts

Integration by parts is a method of integration that involves differentiating one function and integrating the other. It is based on the product rule of differentiation, which states that if $f(x)$ and $g(x)$ are two functions, then the derivative of their product is given by:

$$\frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$$

To apply integration by parts, we need to choose two functions $u$ and $v$ such that $u' = v$ and $v' = -u$. Then, we can use the formula:

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

Choosing the Right Expression for $u$

In the given integral $\int x^3 e^{2x} \, dx$, we need to choose the right expression for $u$. The expression for $u$ should be chosen such that it is easy to differentiate and integrate. In this case, we can choose $u = x^3$ or $u = e^{2x}$.

Choosing $u = x^3$

If we choose $u = x^3$, then we need to find the derivative of $u$, which is $u' = 3x^2$. We also need to find the integral of $v$, which is $\int e^{2x} \, dx$. The integral of $e^{2x}$ is given by:

$$\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C$$

where $C$ is the constant of integration.

Choosing $u = e^{2x}$

If we choose $u = e^{2x}$, then we need to find the derivative of $u$, which is $u' = 2e^{2x}$. We also need to find the integral of $v$, which is $\int x^3 \, dx$. The integral of $x^3$ is given by:

$$\int x^3 \, dx = \frac{x^4}{4} + C$$

where $C$ is the constant of integration.

Conclusion

In conclusion, choosing the right expression for $u$ is crucial in applying integration by parts. In the given integral $\int x^3 e^{2x} \, dx$, we can choose either $u = x^3$ or $u = e^{2x}$. However, the choice of $u$ depends on the ease of differentiation and integration. By carefully choosing the right expression for $u$, we can successfully apply integration by parts and solve the given integral.

Choosing the Right Expression for $u$ in Different Integrals

The choice of $u$ depends on the type of integral and the ease of differentiation and integration. Here are some general guidelines for choosing the right expression for $u$ in different integrals:

• Trigonometric integrals: Choose $u$ to be a trigonometric function such as $\sin x$ or $\cos x$.
• Exponential integrals: Choose $u$ to be an exponential function such as $e^x$ or $e^{-x}$.
• Polynomial integrals: Choose $u$ to be a polynomial function such as $x^n$ or $x^{-n}$.
• Rational integrals: Choose $u$ to be a rational function such as $\frac{1}{x}$ or $\frac{1}{x^2}$.

Examples of Choosing the Right Expression for $u$

Here are some examples of choosing the right expression for $u$ in different integrals:

• Example 1: $\int x^2 \sin x \, dx$
  • Choose $u = x^2$
  • Find the derivative of $u$, which is $u' = 2x$
  • Find the integral of $v$, which is $\int \sin x \, dx = -\cos x + C$
  • Apply integration by parts: $\int x^2 \sin x \, dx = -x^2 \cos x + 2 \int x \cos x \, dx$
• Example 2: $\int e^x \cos x \, dx$
  • Choose $u = e^x$
  • Find the derivative of $u$, which is $u' = e^x$
  • Find the integral of $v$, which is $\int \cos x \, dx = \sin x + C$
  • Apply integration by parts: $\int e^x \cos x \, dx = e^x \sin x - \int e^x \sin x \, dx$

Conclusion

In conclusion, choosing the right expression for $u$ is crucial in applying integration by parts. By carefully choosing the right expression for $u$, we can successfully apply integration by parts and solve different types of integrals. The choice of $u$ depends on the type of integral and the ease of differentiation and integration.