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A short procedure. Let us take f ( x ) = 1 + ( x − 1 ) 3 . Notice that now the integrand becomes f ( x ) + f − 1 ( x ) − 1 [where f − 1 ( x ) is the inverse of f ( x ) ]. We have f ( 1 ) = 1 and f ( 3 ) = 3 . Also note that area bounded by f with x -axis is same as area bounded by f − 1 with y -axis. So, the integral of f + f − 1 between the limits a to b is b f ( b ) − a f ( a ) . In the given question, using this result, we get the value of integral is 3 f ( 3 ) − 1 f ( 1 ) − 2 = 9 − 1 − 2 = 6 . I understand that the explanation is not very clear, but hope it helps.