Math Problem Statement
∫x^3*e^x^2 dx
Solution
To solve the integral , we can use the method of substitution.
Step 1: Substitution
Let: Then, differentiate both sides with respect to :
Step 2: Rewrite the integral
Now, rewrite the original integral in terms of . Since , we substitute and :
Step 3: Use integration by parts
Now, apply integration by parts. Let: Differentiate and integrate: Now, apply the formula for integration by parts: Substituting the values:
Step 4: Substitute back in terms of
Recall that , so:
Final Answer:
Thus, the solution to the integral is: where is the constant of integration.
Would you like more details or have any questions?
Here are 5 related questions for practice:
- How would you solve ?
- Can you apply integration by parts to ?
- What is the derivative of ?
- How do you handle integrals involving without a polynomial?
- What is the solution to ?
Tip: When using substitution, always check if the new variables simplify the integral and avoid unnecessary complexity.
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Math Problem Analysis
Mathematical Concepts
Integration
Substitution Method
Integration by Parts
Exponential Functions
Formulas
Substitution: u = x^2, du = 2x dx
Integration by Parts: ∫v dw = v * w - ∫w dv
Theorems
Integration by Parts Theorem
Substitution Rule in Integration
Suitable Grade Level
Undergraduate Level (Calculus II)