Questions: ∫(x+5)^6(x+4) dx=

Solution Steps

To solve the integral \(\int (x+5)^6 (x+4) \, dx\), we can use the method of substitution. Let's set \(u = x + 5\), which simplifies the integral.

1. Set \(u = x + 5\).
2. Then, \(du = dx\) and \(x = u - 5\).
3. Substitute \(u\) and \(du\) into the integral.
4. Integrate with respect to \(u\).
5. Substitute back \(x\) to get the final answer.

We start by defining the substitution \( u = x + 5 \). This simplifies the integral by transforming the variable \( x \) into \( u \).

Given \( u = x + 5 \), we have \( du = dx \) and \( x = u - 5 \). Substituting these into the integral, we get: \[ \int (u)^6 (u - 1) \, du \]

Next, we expand the integrand: \[ \int u^6 (u - 1) \, du = \int (u^7 - u^6) \, du \]

We integrate each term separately: \[ \int u^7 \, du - \int u^6 \, du = \frac{u^8}{8} - \frac{u^7}{7} + C \]

Finally, we substitute back \( u = x + 5 \) to express the integral in terms of \( x \): \[ \frac{(x + 5)^8}{8} - \frac{(x + 5)^7}{7} + C \]

Final Answer