Abstract In this paper, we investigate and evaluate the analytical expressions for some definite integrals of Srinivasa Ramanujan in terms of Meijer’s G-function by using the Laplace transforms of sin ⁡ ( β ⁢ x 2 ) {\sin(β x^{2})} , cos ⁡ ( β ⁢ x 2 ) {\cos(β x^{2})} , x ⁢ sin ⁡ ( β ⁢ x 2 ) {x\sin(β x^{2})} and x ⁢ cos ⁡ ( β ⁢ x 2 ) {x\cos(β x^{2})} . In addition, we investigate a number of infinite summation formulas involving Meijer’s G-function and closed-form evaluation of some related infinite series.