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Integration by parts: logarithms, powers and a returning integral

Find the following integrals:

(a) ∫x2ln⁡x dx\displaystyle\int x^2\ln x\,dx

(b) ∫ln⁡(x2+1) dx\displaystyle\int \ln(x^2+1)\,dx

(c) ∫xtan⁡−1(x2) dx\displaystyle\int x\tan^{-1}(x^2)\,dx

(d) ∫x2ex dx\displaystyle\int x^2e^x\,dx

(e) ∫excos⁡x dx\displaystyle\int e^x\cos x\,dx

(f) ∫θsin⁡θcos⁡θ dθ\displaystyle\int \theta\sin\theta\cos\theta\,d\theta.

CJC · Discussion · Question 8

Choose u so that differentiation simplifies it; keep the returning integral on both sides when parts cycles.

Read the worked solution