Problems on Techniques of Integration

Use the Integration by Parts. Set

$$\left\{\begin{array}{lll} u &=&x\\ dv &=& \cos(x)dx\;. \end{array}\right.$$

Then

$$\left\{\begin{array}{lll} du &=&dx\\ v &=& \sin(x)\;. \end{array}\right.$$

So

$$\int x \cos(x) dx = x \sin(x) - \int sin(x) dx$$

or

$$\int x \cos(x) dx = x \sin(x) - [-\cos(x)] + C = x \sin(x) + \cos(x) + C\;.$$

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