Evaluating ∫ x·cos(x²) dx

Real Analysis · Axiom Academy

Work through this integral step by step with u-substitution, spotting the inner function whose derivative is already present. Evaluate the indefinite integral using u-substitution. Notice that the derivative of the inner function x^2 is 2x , and a factor of x already sits out front — that is the signal to substitute. The substitution u = x^2 converts the awkward integrand into a plain cosine in u — a standard integral. Nice work — you evaluated end to end with u-substitution. Here is what carries over to the next integral: Spot the inner function: pick u so that its derivative already appears (up to a constant) in the integrand — here u = x^2 , and 2x is right there. Convert the differential: from u = x^2 we get , so — that stray rides along through the whole integral. Integrate, then substitute back: , and replacing u = x^2 gives the answer in x . Always check: differentiate the answer — , which is the original integrand. U-substitution is the reverse of the chain rule. Train your eye to recognize a composite function paired with (a constant times) its inner derivative, and integrals like this become routine.

This is the written version of the interactive lesson above. See the full Real Analysis course.