Integrate y = x^3. Use the formula: Integral x^n = (x^n+1)/n+1 where, for x^3, n is 3 and n +1 is 4. As a consequence, the integral x^3 = (x^4)/4.
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Differentiate (x^4)/4. Use the formula dy/dx = nx^n-1 where, for (x^4)/4, n is 4 and n-1 is 3. As a consequence, dy/dx = (4x^3/)4, which, as the 4 cancels, leaves x^3.
Compare the results of step 2 with the original equation. From this example, the original equation was y = x^3 and the derivative of the integral of y= x^3, or x^4/4, is also y = x^3.