Math Problem Statement

If ∫ f ( x ) d x

( x + 3 ) 4 e x + ln ( | x | ) + C , what is f ( x ) ?

Solution

To determine f(x)f(x), we differentiate the given integral expression:

f(x)dx=(x+3)4ex+ln(x)+C\int f(x) \, dx = (x+3)^4 e^x + \ln(|x|) + C

Differentiating both sides with respect to xx:

f(x)=ddx[(x+3)4ex+ln(x)+C]f(x) = \frac{d}{dx} \left[ (x+3)^4 e^x + \ln(|x|) + C \right]

Step-by-Step Differentiation

  1. Differentiate (x+3)4ex(x+3)^4 e^x: Using the product rule, ddx[uv]=uv+uv\frac{d}{dx}[u v] = u'v + uv', where u=(x+3)4u = (x+3)^4 and v=exv = e^x: [ \frac{d}{dx} \left[ (x+3)^4 e^x \right] = \frac{d}{dx}[(x+3)^4] \cdot e^x + (x+3)^4 \cdot \frac{d}{dx}[e^x] ]

    • ddx[(x+3)4]=4(x+3)3\frac{d}{dx}[(x+3)^4] = 4(x+3)^3
    • ddx[ex]=ex\frac{d}{dx}[e^x] = e^x

    Substituting: ddx[(x+3)4ex]=4(x+3)3ex+(x+3)4ex\frac{d}{dx} \left[ (x+3)^4 e^x \right] = 4(x+3)^3 e^x + (x+3)^4 e^x

    Combine terms: [ \frac{d}{dx} \left[ (x+3)^4 e^x \right] = e^x \left[ 4(x+3)^3 + (x+3)^4 \right] ]

  2. Differentiate ln(x)\ln(|x|): Recall ddx[ln(x)]=1x\frac{d}{dx}[\ln(|x|)] = \frac{1}{x}: ddx[ln(x)]=1x\frac{d}{dx}[\ln(|x|)] = \frac{1}{x}

  3. Differentiate the constant CC: The derivative of a constant is zero: ddx[C]=0\frac{d}{dx}[C] = 0

Combine Results

Adding all these results gives: f(x)=ex[4(x+3)3+(x+3)4]+1xf(x) = e^x \left[ 4(x+3)^3 + (x+3)^4 \right] + \frac{1}{x}

Simplify the expression: f(x)=ex(x+3)3[4+(x+3)]+1xf(x) = e^x (x+3)^3 [4 + (x+3)] + \frac{1}{x}

f(x)=ex(x+3)3(x+7)+1xf(x) = e^x (x+3)^3 (x+7) + \frac{1}{x}

Final Answer:

f(x)=ex(x+3)3(x+7)+1xf(x) = e^x (x+3)^3 (x+7) + \frac{1}{x}

Would you like a detailed explanation of any specific step?

Related Questions:

  1. What is the significance of logarithmic differentiation in solving such problems?
  2. How does the product rule apply to expressions involving both polynomials and exponentials?
  3. Can this function f(x)f(x) have singularities? If so, where?
  4. How can this method be extended to solve f(x)dx\int f(x) dx for a general f(x)f(x)?
  5. What happens if ln(x)\ln(|x|) is replaced with another logarithmic function?

Tip:

When differentiating products or compositions, ensure each rule (e.g., product, chain) is applied consistently to avoid missing terms.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Logarithmic Functions
Exponential Functions
Product Rule

Formulas

Product Rule: d(uv)/dx = u'v + uv'
Derivative of ln(|x|): d/dx[ln(|x|)] = 1/x
Derivative of e^x: d/dx[e^x] = e^x

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Calculus or Advanced High School Calculus