#### To determine

**To find:**

∫03xf(x2)dx

#### Answer

2

#### Explanation

**1) Concept:**

The substitution rule:

If u=g(x) is a differentiable function whose range is I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du.

Here, g(x) is substituted as u and then g’(x)dx =du.

**2) Given:**

The function f is continuous and ∫09f(x)dx=4

**3) calculation:**

To find

∫03xf(x2)dx

use the substitution method.

Substitute x2=u.

Differentiating with respect to x

2xdx=du

xdx=12du

The limits change and the new limits of integration are calculated by substituting

for x=0, u=02=0 & for x=3, u=32 =9

Therefore, the given integral becomes

∫03xf(x2)dx=12∫09f(u)du

We are given that f is continuous and ∫09f(x)dx=4

Therefore,

12∫09f(u)du=12(4)

So it follows that

∫03xf(x2)dx=2

**Conclusion:**

Therefore,

∫03xf(x2)dx=2