Suppose that $f$ is a continuous function such that
$\int_{1}^{2} f (x) d x = 4$
Determine the value of the definite integral
$\int_{1}^{4} \frac{f (\sqrt{x})}{\sqrt{x}} d x$

Answer

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Answer

So, the value of the definite integral $\int_{1}^{4} \frac{f (\sqrt{x})}{\sqrt{x}} d x$ is $8$ .

Steps

Step 1 ：Let's use the substitution method to solve the integral. We can let $u = \sqrt{x}$ , then $d u = \frac{1}{2 \sqrt{x}} d x$ . This changes the limits of integration from 1 to 2, and the integral becomes $\int_{1}^{2} 2 f (u) d u$ .

Step 2 ：We know that $\int_{1}^{2} f (x) d x = 4$ , we can substitute this into the integral to find the value.

Step 3 ：So, the value of the definite integral $\int_{1}^{4} \frac{f (\sqrt{x})}{\sqrt{x}} d x$ is $8$ .

Suppose that f is a continuous function such that∫12f(x)dx=4Determine the value of the definite integral∫14f(x)xdx

Answer

Popular Problems

Suppose that $f$ is a continuous function such that
$\int_{1}^{2} f (x) d x = 4$
Determine the value of the definite integral
$\int_{1}^{4} \frac{f (\sqrt{x})}{\sqrt{x}} d x$