First make a substitution and then use integration by parts to evaluate the integral. x³cos(x²)dx

Solution:

Given f(x) = x³cos(x²)dx

We cannot solve this integral directly hence, we use integration by substitution method

Let x² = t

2xdx = dt

xdx = (1/2)dt

f(x) = ∫ x³cos(x²) dx

f(x) = ∫ x²cos(x²) × x dx

f(x)= ∫ tcostdt/2

f(x)= 1/2 ∫ tcostdt

Let u = t

du = 1dt

v = sint

dv = costdt

By using integral by parts, we have

∫ uv = u∫v - ∫u’∫vdu

f(x) = 1/2 ∫ tcostdt = 1/2 {t ∫sint - ∫1.sintdt }

We have t = x², 1.dt = 2xdx

f(x) = 1/2 x²sin(x²) - xcos(x²)dx

Integrate by substitution again to finish.

∫x³cos(x²)dx = 1/2x²sin(x²) + 1/2cos(x²) + C

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First make a substitution and then use integration by parts to evaluate the integral. x³cos(x²)dx

Summary:

By substitution and evaluating, we get ∫x³cos(x²)dx = 1/2x²sin(x²) + 1/2cos(x²) + C