Solving ∫x²sin(x)dx

Calculus 2 · Axiom Academy

Work through double integration by parts step by step, tracking signs carefully to reach the answer. Evaluate the indefinite integral . The integrand multiplies a polynomial by a trig function, so integration by parts is the tool — and we will need it twice. Nice work — you carried a double integration by parts all the way through. Here is what to hold onto: LIATE for u: pick u by priority — Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Here x^2 (algebraic) outranks (trig). Repeat when needed: if the first pass leaves another integral ( ), apply integration by parts again — the polynomial degree drops each time. Mind the signs: the usual slip is a sign error. Watch the subtraction in the formula and double negatives like . Track u, du, v, and dv deliberately at each stage and these multi-step integrals become routine.

This is the written version of the interactive lesson above. See the full Calculus 2 course.