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 $\int_{1}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} dx$ is $\boxed{8}$.

Steps

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

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

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

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

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
\[
\int_{1}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} d x
\]