First make a substitution and then use integration by parts to evaluate the integral.

Question

x^3cos(x^2)dx ? Thanks for your help :)

Answer 1

substitute the argument of the cosine:
t = x²  ----> x = √t
=> dx = 1/(2∙√t) dx

∫ x³ ∙ cos(x²) dx
= ∫ (√t)³ ∙ cos(t) / (2∙√t) dt
= ∫ ((√t)²/2) ∙ cos(t) dt
= (1/2) ∙ ∫ t∙cos(t) dt

integrate by parts with
u = t => u' = 1
v' = cos(t) => v = sin(t)

= (1/2) ∙ [ t∙sin(t) - ∫ sin(t) dt ]
= (1/2) ∙ [ t∙sin(t) + cos(t) ] + c
= (1/2) ∙ [ x²∙sin(x²) + cos(x²) ] + c

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