To calculate the integral of the function \(f(x) = 3x^2 - 5x + \sqrt{2}\) from \(x = 0.512\) to \(x = 3.721\), we first need to find the antiderivative of the function and then evaluate it at the upper and lower limits of integration.

The antiderivative of \(f(x)\) is:
\[
F(x) = \int (3x^2 - 5x + \sqrt{2}) \, dx = \int 3x^2 \, dx - \int 5x \, dx + \int \sqrt{2} \, dx
\]

Integrating term by term, we get:
\[
F(x) = \frac{3}{3}x^3 - \frac{5}{2}x^2 + \sqrt{2}x + C
\]
\[
F(x) = x^3 - \frac{5}{2}x^2 + \sqrt{2}x + C
\]

Since we are evaluating a definite integral, we do not need to worry about the constant of integration \(C\). Now, we evaluate the antiderivative at the upper and lower limits:

\[
F(3.721) = (3.721)^3 - \frac{5}{2}(3.721)^2 + \sqrt{2}(3.721)
\]
\[
F(0.512) = (0.512)^3 - \frac{5}{2}(0.512)^2 + \sqrt{2}(0.512)
\]

The definite integral is then:
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx = F(3.721) - F(0.512)
\]

Now, let's calculate the values:

\[
F(3.721) = (3.721)^3 - \frac{5}{2}(3.721)^2 + \sqrt{2}(3.721)
\]
\[
F(3.721) = 51.421961 - \frac{5}{2}(13.845841) + \sqrt{2}(3.721)
\]
\[
F(3.721) = 51.421961 - 34.6146025 + 5.260254
\]
\[
F(3.721) = 51.421961 - 34.6146025 + 5.260254 \approx 22.0676125
\]

\[
F(0.512) = (0.512)^3 - \frac{5}{2}(0.512)^2 + \sqrt{2}(0.512)
\]
\[
F(0.512) = 0.134217728 - \frac{5}{2}(0.262144) + \sqrt{2}(0.512)
\]
\[
F(0.512) = 0.134217728 - 0.65536 + 0.7236067977
\]
\[
F(0.512) = 0.134217728 - 0.65536 + 0.7236067977 \approx 0.202464526
\]

Finally, the definite integral is:
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx = F(3.721) - F(0.512)
\]
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx = 22.0676125 - 0.202464526
\]
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx \approx 21.865148
\]

So, the integral of \(3x^2 - 5x + \sqrt{2}\) from \(x = 0.512\) to \(x = 3.721\) is approximately \(21.865148\).
x = 0.512\)에서 \(x = 3.721\)까지의 함수 \(f(x) = 3x^2 - 5x + \sqrt{2}\)의 적분을 계산하려면 먼저 함수의 역함수를 구한 다음 적분의 상하한에서 평가해야 합니다.

f(x)의 역함수는 다음과 같습니다:
\[
F(x) = \int (3x^2 - 5x + \sqrt{2}) \, dx = \int 3x^2 \, dx - \int 5x \, dx + \int \sqrt{2} \, dx
\]

항을 적분하면 다음과 같습니다:
\[
F(x) = \frac{3}{3}x^3 - \frac{5}{2}x^2 + \sqrt{2}x + C
\]
\[
F(x) = x^3 - \frac{5}{2}x^2 + \sqrt{2}x + C
\]

정적분을 평가하므로 적분 상수 \(C\)에 대해 걱정할 필요가 없습니다. 이제 상하한에서 역함수를 평가해 보겠습니다:

\[
F(3.721) = (3.721)^3 - \frac{5}{2}(3.721)^2 + \sqrt{2}(3.721)
\]
\[
F(0.512) = (0.512)^3 - \frac{5}{2}(0.512)^2 + \sqrt{2}(0.512)
\]

그러면 정적분은 다음과 같습니다:
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx = F(3.721) - F(0.512)
\]

이제 값을 계산해 보겠습니다:

\[
F(3.721) = (3.721)^3 - \frac{5}{2}(3.721)^2 + \sqrt{2}(3.721)
\]
\[
F(3.721) = 51.421961 - \frac{5}{2}(13.845841) + \sqrt{2}(3.721)
\]
\[
F(3.721) = 51.421961 - 34.6146025 + 5.260254
\]
\[
F(3.721) = 51.421961 - 34.6146025 + 5.260254 \approx 22.0676125
\]

\[
F(0.512) = (0.512)^3 - \frac{5}{2}(0.512)^2 + \sqrt{2}(0.512)
\]
\[
F(0.512) = 0.134217728 - \frac{5}{2}(0.262144) + \sqrt{2}(0.512)
\]
\[
F(0.512) = 0.134217728 - 0.65536 + 0.7236067977
\]
\[
F(0.512) = 0.134217728 - 0.65536 + 0.7236067977 \approx 0.202464526
\]

마지막으로, 정적분은 다음과 같습니다:
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx = F(3.721) - F(0.512)
\]
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx = 22.0676125 - 0.202464526
\]
\[
\int_{0.512}^{3.721} (3x^2 - 5x + \sqrt{2}) \, dx \approx 21.865148
\]

따라서 \(x = 0.512\)에서 \(x = 3.721\)까지의 \(3x^2 - 5x + \sqrt{2}\)의 적분은 약 21.865148입니다.