5.2 Rules for integration

5.2.1 Sum rule

Example 5.5: 

Find {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}(\mathop{sin}\nolimits x +\mathop{ cos}\nolimits x)dx.

Solution: 

\eqalignno{ {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}(\mathop{sin}\nolimits x +\mathop{ cos}\nolimits x)dx ={\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}\mathop{ sin}\nolimits x\kern 1.66702pt dx +{\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}\mathop{ cos}\nolimits x\kern 1.66702pt dx ={ \left [−\mathop{cos}\nolimits (x) +\mathop{ sin}\nolimits (x)\right ]}_{ 0}^{π} = −(−1) − (−1) = 2 & & }

5.2.2 Constant multiple

Example 5.6: 

Find \mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits \sqrt{x}({x}^{3} − 3)\kern 1.66702pt dx.

Solution: 

Expand the integrand;

\eqalignno{ \mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits \sqrt{ x}({x}^{3} − 3)\kern 1.66702pt dx =\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits ({x}^{7∕2} − 3{x}^{1∕2})\kern 1.66702pt dx =\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits {x}^{7∕2}\kern 1.66702pt dx − 3\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits {x}^{1∕2}\kern 1.66702pt dx = {2\over 9}{x}^{9∕2} − 3{2\over 3}{x}^{3∕2} + c = {2\over 9}{x}^{9∕2} − 2{x}^{3∕2} + c.&& }