Question

Find the antiderivative. \(\int \frac{d x}{(x+4)^{3}}\)

Short answer

The antiderivative is \(-\frac{1}{2(x+4)^2} + C\).

Step-by-step solution

Identify the Integral Type

Recognize that the given integral \( \int \frac{d x}{(x+4)^{3}} \) is a basic power of binomials integral that can be simplified using a substitution method.

Apply Substitution

Set the substitution \( u = x + 4 \). Then the derivative \( du = dx \). This allows us to rewrite the integral in terms of \( u \): \( \int \frac{1}{u^3} \, du \).

Rewrite with Exponent Notation

The integral \( \int \frac{1}{u^3} \, du \) is the same as \( \int u^{-3} \, du \). This simplifies the problem for integration.

Integrate using Power Rule

Apply the power rule for integration, which states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), for \( n eq -1 \). Here, \( n = -3 \): \[ \int u^{-3} \, du = \frac{u^{-2}}{-2} + C = -\frac{1}{2u^2} + C \].

Substitute Back Original Variable

Since \( u = x + 4 \), substitute back \( u \) with \( x + 4 \) into the antiderivative. This yields: \(-\frac{1}{2(x+4)^2} + C\).

Key concepts

Substitution Method

The substitution method is a powerful technique for solving integrals when faced with a complex function. It involves changing variables to simplify the integral into a more manageable form. Here, the idea is to substitute a part of the integral with a new variable, typically denoted by \( u \).

• The first step is to identify the part of the integral that can be substituted. In the example \( \int \frac{d x}{(x+4)^{3}} \), we choose \( x + 4 \) to substitute.
• Set \( u = x + 4 \). This transformation simplifies the expression.
• Next, find the differential of the new variable: \( du = dx \). This helps in changing the integration variable from \( x \) to \( u \).
• Replace \( x + 4 \) in the integral with \( u \), leading to the simpler integral \( \int \frac{1}{u^3} \, du \).

By doing this substitution, the problem becomes much easier to handle as it moves from a complex binomial form to a simpler power function. As you will see, this step lays the groundwork for applying the power rule of integration.

Integration by Parts

While this specific exercise was solved using the substitution method, the technique of integration by parts is another handy method often employed for integrals. Integration by parts is used when the antiderivative of a product of functions is needed. The formula for integration by parts is:\[\int u \, dv = uv - \int v \, du\]Here, you need to:

• Choose two parts of the integral where one part \( u \) will be differentiated, and another \( dv \) will be integrated.
• Compute \( du \) as the derivative of \( u \) and integrate \( dv \) to find \( v \).

The choice of \( u \) and \( dv \) is crucial for simplifying the calculation. Typically, the function \( u \) should be one that simplifies when differentiated, whereas \( dv \) can be easily integrated. This approach continuously reduces the complexity of the integral by simplifying one term at a time.

Power Rule for Integration

The power rule for integration is a fundamental and straightforward concept for finding antiderivatives, especially when dealing with polynomial functions or simple powers of \( u \).The power rule states that for any real number \( n \) except \( n = -1 \), the integral of \( u^n \) is:\[\int u^n \, du = \frac{u^{n+1}}{n+1} + C\]This exercise applied the power rule once the substitution was made: \( \int u^{-3} \, du \). Here, \( n = -3 \), which is valid.

• Increase the exponent by 1: \(-3 + 1 = -2 \).
• Divide the term by the new exponent: \( \frac{u^{-2}}{-2} \).

Simplifying gives \( -\frac{1}{2u^2} + C \). Finally, substitute back to the original variable \( x \), converting \( u \) back to \( x+4 \), resulting in the solution \( -\frac{1}{2(x+4)^2} + C \). This simple rule is powerful for efficiently solving integrals that appear in polynomial or power form.