More Integration Procedures, 9

Category: Integral Calculus, Trigonometry

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Evaluate

Solution:

Consider the given equation above

There are two functions in the given equation which are x³ and sin x, respectively. If

 then

If

then

By applying the integration by parts, we have

Since the second term at the right side of the equation have two functions which are x² and cos x, then we have to apply the integration by parts again. If

then

If

then

Again, by applying the integration by parts, we have

Therefore, the first integration by parts becomes

Since the second term at the right side of the equation have two functions which are x and sin x, then we have to apply the integration by parts again. If

then

If

then

Again, by applying the integration by parts, we have

 

 

Therefore, the final answer is