Find:
Solution (a) Notice that the numerator equals the derivative of the denominator, except for a constant factor. So, we try substitution. Let u=4x3−1. Then du=12x2dx so 5x2dx=512du. ∫5x2dx4x3−1=∫512duu=512∫duu=512ln|u|+C=512ln|4x3−1|+C
(b) Here, the numerator equals the derivative of the denominator. So, we use the substitution u=ex+4. Then du=exdx. ∫exex+4dx=∫1ex+4⋅exdx=∫1udu=ln|u|+C=↑u=ex+4>0ln(ex+4)+C