Question

1. the function ( f ) is continuous and ( int_{4}^{19} f(u) , du = 10 ). what is the value of ( int_{1}^{4} left x cdot f(x^2 + 3)

ight dx )

(a) ( \frac{5}{2} )

(b) ( 5 )

(c) ( 10 )

(d) ( 20 )

(e) ( 40 )

Explanation:

Step1: Use substitution method

Let \( u = x^2 + 3 \), then \( du = 2x \, dx \), so \( \frac{1}{2} du = x \, dx \).

Step2: Find new limits of integration

When \( x = 1 \), \( u = 1^2 + 3 = 4 \).
When \( x = 4 \), \( u = 4^2 + 3 = 19 \).

Step3: Rewrite the integral

The integral \( \int_{1}^{4} [x \cdot f(x^2 + 3)] dx \) becomes \( \frac{1}{2} \int_{4}^{19} f(u) du \).

Step4: Use the given integral value

We know that \( \int_{4}^{19} f(u) du = 10 \), so substitute this value into the rewritten integral: \( \frac{1}{2} \times 10 = 5 \).

Answer:

B. 5