Math Problem Statement

Déterminer toutes les primitives des fonctions suivantes : f(x) = e^(3x) / (1 + e^(3x)), f(x) = ln(x)/x, f(x) = cos(x)sin^2(x), etc.

Solution

The exercise asks to find the antiderivatives (or primitives) of the given functions. Let's go through some of them:

  1. f(x)=e3x1+e3xf(x) = \frac{e^{3x}}{1 + e^{3x}}

To solve this, we can use a substitution: Let u=1+e3xu = 1 + e^{3x}, then du=3e3xdxdu = 3e^{3x} dx. The result will lead to an expression involving the natural logarithm.

  1. f(x)=ln(x)xf(x) = \frac{\ln(x)}{x}

This is a standard form. We know that the antiderivative of ln(x)x\frac{\ln(x)}{x} is ln2(x)2\frac{\ln^2(x)}{2}, because by applying integration by parts, we can solve this.

  1. f(x)=cos(x)sin2(x)f(x) = \cos(x) \sin^2(x)

This can be simplified using the identity sin2(x)=1cos2(x)\sin^2(x) = 1 - \cos^2(x), and then solved by basic trigonometric integration techniques.

  1. f(x)=1xln(x)f(x) = \frac{1}{x \ln(x)}

This requires the substitution u=ln(x)u = \ln(x), leading to a result involving the natural logarithm.


Would you like me to continue with specific steps for any of the functions or clarify one in detail? Here are 5 follow-up questions to extend the current discussion:

  1. Would you like a detailed step-by-step solution for problem 1?
  2. Do you want to explore the method of substitution in more detail?
  3. Should we focus on a specific integration technique like parts or trigonometric identities?
  4. Are there any particular integrals from the list you'd like prioritized?
  5. Would it be helpful to review general strategies for tackling integrals?

Tip: Always try substitution first when dealing with logarithmic and exponential functions.

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution
Trigonometric Integrals
Logarithmic Integrals
Integration by Parts

Formulas

∫e^(3x)dx
∫(ln(x)/x)dx
Trigonometric identity: sin^2(x) = 1 - cos^2(x)
Integration by parts formula: ∫u dv = uv - ∫v du

Theorems

Fundamental Theorem of Calculus
Substitution Theorem
Integration by Parts Theorem

Suitable Grade Level

Undergraduate Calculus