Evaluate the integral \[ \int x^{3}\left(x^{4}-9\right)^{26} d x \] by making the substitution $u=x^{4}-9$. NOTE: Your answer should be in terms of $x$ and not $u$.
Solution
Step 1 ：Let's start by making the substitution $u = x^4 - 9$. This means $du = 4x^3 dx$. However, our integral is in terms of $x^3 dx$, not $4x^3 dx$. So we'll need to adjust our $du$ to match the integral. We can do this by dividing both sides of $du = 4x^3 dx$ by 4, giving us $du/4 = x^3 dx$.
Step 2 ：Now we can substitute $u$ and $du/4$ into our integral, and then solve. The integral becomes quite complex and involves high powers of x.
Step 3 ：Finally, we can simplify it by substituting back the original $u = x^4 - 9$ into the integral. The final answer is \(\boxed{x^{108}/108 - 9x^{104}/4 + 1053x^{100}/4 - 78975x^{96}/4 + 2132325x^{92}/2 - 88278255x^{88}/2 + 2913182415x^{84}/2 - 78655925205x^{80}/2 + 3539516634225x^{76}/4 - 67250816050275x^{72}/4 + 1089463220014455x^{68}/4 - 15153442969291965x^{64}/4 + 45460328907875895x^{60} - 472088030966403525x^{56} + 4248792278697631725x^{52} - 33140579773841527455x^{48} + 894795653893721241285x^{44}/4 - 5210868807969317816895x^{40}/4 + 26054344039846589084475x^{36}/4 - 111073782485661774518025x^{32}/4 + 199932808474191194132445x^{28}/2 - 599798425422573582397335x^{24}/2 + 1472232498764498793157095x^{20}/2 - 2880454888887062856176925x^{16}/2 + 8641364666661188568530775x^{12}/4 - 9332673839994083654013237x^{8}/4 + 6461081889226673298932241x^{4}/4}\)
