Question

\(\int(x+4)(x+3)^{7} d x=\) (a) \(\left[(x+3)^{9} / 9\right]-\left[(x+3)^{8} / 8\right]\) (b) \(\left[\left\\{(x+3)^{8}(8 x+33)\right\\} / 72\right]\) (c) \(\left[\left\\{(x+3)^{8}(8 x+33)\right\\} / 72\right]\) (d) \(\left[(x+3)^{8} / 8\right]\)

Short answer

The short answer is: \(\left[(x+3)^{9} / 9\right]-\left[(x+3)^{8} / 8\right]\)

Step-by-step solution

Apply Substitution

Let u be x + 3. So, our integral becomes: \[ \int(u-1)u^7 du \]

Expand the Expression

Now, expand the expression by distribution: \[ \int(u^8 - u^7) du \]

Integrate with respect to u

To integrate, we apply the power rule, which states that the integral of \(u^n\) is \(\frac{u^{n+1}}{n+1}\). So, we have: \[ \int(u^8 - u^7) du = \frac{u^9}{9} - \frac{u^8}{8} + C \] Here, C represents the constant of integration.

Substitute back in terms of x

Replace u with x + 3: \[ \frac{(x+3)^9}{9} - \frac{(x+3)^8}{8} + C \] The correct answer is (a) \(\left[(x+3)^{9} / 9\right]-\left[(x+3)^{8} / 8\right]\).

Key concepts

Substitution Method

The substitution method is a powerful tool in calculus used to simplify the process of integration. It involves changing variables to make the integral easier to solve, which can be extremely helpful when dealing with complex expressions.
When using substitution, the general idea is to transform the variable into a simpler form. For instance, if you have \( \int f(g(x)) \, g'(x) \, dx \), you would set \( u = g(x) \). The differential \( du \) is then \( g'(x) dx \. \)

• Choose \( u = x + 3 \).
• Differential \( du = dx \).
• Replace original expression \( (x + 4)(x+3)^7 dx \) with \( (u-1)u^7 du \).

This substitution simplifies the integral significantly by reducing it to \( \int (u^8 - u^7) \, du \). This expression is much easier to integrate using simpler rules.

Power Rule in Integration

The power rule is perhaps one of the most fundamental rules for integration. It helps you find the integral of polynomials quickly and accurately.
According to the power rule, if you need to integrate \( x^n \), the formula to use is \( \frac{x^{n+1}}{n+1} + C \. \)
In our previous substitution method example, after expanding \( (u-1)u^7 \) to \( u^8 - u^7 \, \) we apply the power rule:

• For \( u^8 \, \int u^8 \, du = \frac{u^{9}}{9} \).
• For \( u^7 \, \int u^7 \, du = \frac{u^{8}}{8} \).

The power rule allows you to easily integrate terms like these by increasing the exponent by one, and then dividing by the new exponent. The constant \( C \) signifies any constant factor that was originally part of the function.

Polynomial Integration

Polynomial integration refers to integrating functions that are polynomials. As polynomials can include terms like \( x^2, x^3, \) and others, integrating them involves integrating each term individually using the power rule.
For the given problem, after substitution and simplification to obtain \( u^8 - u^7 \, \) we treated this as two separate polynomials:

• \( \int u^8 \quad \text{becomes} \quad \frac{u^{9}}{9} \).
• \( \int u^7 \quad \text{becomes} \quad \frac{u^{8}}{8} \).

The simplicity of polynomial integration comes from applying the power rule term by term.
Each polynomial term in the expression gets its own separate integration. Finally, you add any necessary constants for indefinite integrals.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is crucial in calculus.

• Indefinite integrals represent a family of functions and include a constant of integration, \( C \, \) since the antiderivative can vary by a constant.
• Definite integrals, on the other hand, evaluate to a specific number and are used to find the area under a curve between two limits.
  They do not include the constant \( C \) because the integral's limits replace it. \

In the provided problem, we find an indefinite integral because there's no specified upper or lower limit.
This results in the expression \( \frac{(x+3)^9}{9} - \frac{(x+3)^8}{8} + C \).
Had it been a definite integral, specific values would replace the limits of integration, yielding a numeric result.