Math Problem Statement
Solution
The exercise asks to find the antiderivatives (or primitives) of the given functions. Let's go through some of them:
To solve this, we can use a substitution: Let , then . The result will lead to an expression involving the natural logarithm.
This is a standard form. We know that the antiderivative of is , because by applying integration by parts, we can solve this.
This can be simplified using the identity , and then solved by basic trigonometric integration techniques.
This requires the substitution , 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:
- Would you like a detailed step-by-step solution for problem 1?
- Do you want to explore the method of substitution in more detail?
- Should we focus on a specific integration technique like parts or trigonometric identities?
- Are there any particular integrals from the list you'd like prioritized?
- 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
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