This integral is solved by making the substitution u = 2x^{3} + 7 and using the power rule. We find du = 6x^{2} dx. After the substitution, we have ∫ u^{–3/2} du, which is solved using the power rule with n = –3/2.
u = 2x^{3} + 7; du = 6x^{2} dx  



= ∫ u^{–3/2} du  
= –2u^{–1/2} + C  
= –2 (2x^{3} + 7)^{–1/2} + C 