Evaluating ∫x√(x² + 1)dx

Calculus 2 · Axiom Academy

Master the technique of u-substitution through a step-by-step walkthrough Evaluate the indefinite integral using u-substitution. The key is to spot an inner function whose derivative already appears in the integrand. Excellent work! You've successfully solved an integral using u-substitution. Here's what we learned: Choose u wisely: Look for an "inner function" whose derivative appears (or can be made to appear) elsewhere in the integrand. Here, u = x^2 + 1 worked because , and we had in the original integral. Express everything in terms of u: After finding du , solve for dx and substitute both u and dx into the integral. Nothing from the original variable should remain! Integrate in u-space: Once fully converted, the integral should be simpler. Use standard integration techniques (power rule, etc.). Always back-substitute: Replace u with the original expression to get your final answer in terms of x . Don't forget the constant of integration! Definite integrals require special care: When evaluating definite integrals, either change the limits to u-values OR back-substitute first and then apply the original x-limits. Never mix u and x limits! U-substitution is one of the most powerful integration techniques. Practice identifying good substitution candidates, and remember: when you see a function and its derivative (or a constant multiple), think u-substitution!

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