# First make a substitution and then use integration by parts to evaluate the integral. x^{3}cos(x^{2})dx

**Solution:**

Given f(x) = x^{3}cos(x^{2})dx

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

Let x^{2} = t

2xdx = dt

xdx = (1/2)dt

f(x) = ∫ x^{3}cos(x^{2}) dx

f(x) = ∫ x^{2}cos(x^{2}) × 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 = x^{2}, 1.dt = 2xdx

f(x) = 1/2 x^{2}sin(x^{2}) - xcos(x^{2})dx

Integrate by substitution again to finish.

∫x^{3}cos(x^{2})dx = 1/2x^{2}sin(x^{2}) + 1/2cos(x^{2}) + C

## First make a substitution and then use integration by parts to evaluate the integral. x^{3}cos(x^{2})dx

**Summary:**

By substitution and evaluating, we get ∫x^{3}cos(x^{2})dx = 1/2x^{2}sin(x^{2}) + 1/2cos(x^{2}) + C

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