Question
\( \int _{}^{}15(x+2)^{4} d \)
Ask by Norris Ruiz. in Mexico
Feb 27,2025
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Tutor-Verified Answer
Answer
The integral of \(15(x+2)^{4}\) with respect to \(x\) is \(3(x+2)^{5} + C\), where \(C\) is a constant.
Solution
Calculate the integral \( \int 15(x+2)^{4} d(x) \).
Evaluate the integral by following steps:
- step0: Solution:
\(\int 15\left(x+2\right)^{4} d(\times x\)
- step1: Evaluate the integral:
\(15\left(x+2\right)^{4}\left(\right)x\)
- step2: Multiply the expression:
\(15\left(\right)x\left(x+2\right)^{4}\)
- step3: Add the constant of integral C:
\(15\left(\right)x\left(x+2\right)^{4} + C, C \in \mathbb{R}\)
The integral of \(15(x+2)^{4}\) with respect to \(x\) is \(15\frac{(x+2)^{5}}{5} + C\), where \(C\) is an arbitrary constant.
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Simplify this solution Bonus Knowledge
To solve the integral \( \int 15(x+2)^{4} \, dx \), we can use the power rule for integration. First, we factor out the constant: \[ 15 \int (x+2)^{4} \, dx \] Now, we apply the power rule, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). In our case, we substitute \( u = x + 2 \), so when we differentiate \( u \) we get \( du = dx \). Thus, the integral becomes: \[ 15 \cdot \frac{(u)^{5}}{5} + C = 3u^{5} + C \] Substituting back for \( u \): \[ 3(x+2)^{5} + C \] So the final answer is: \[ 3(x+2)^{5} + C \]
