Questions: Find the indefinite integral. ∫ sqrt(1+2 x^4)(8 x^3) dx ∫ sqrt(1+2 x^4)(8 x^3) dx=

Transcript text: Find the indefinite integral. \[ \begin{array}{r} \int \sqrt{1+2 x^{4}}\left(8 x^{3}\right) d x \\ \int \sqrt{1+2 x^{4}}\left(8 x^{3}\right) d x= \end{array} \]

Solution

Solution Steps

To solve the indefinite integral \(\int \sqrt{1+2x^4}(8x^3) \, dx\), we can use a substitution method. We will set \(u = 1 + 2x^4\), which simplifies the integral. Then, we find \(du\) in terms of \(dx\) and substitute back into the integral to solve it.

Step 1: Set Up the Integral

We start with the indefinite integral: \[ \int \sqrt{1 + 2x^4} \cdot (8x^3) \, dx \]

Step 2: Perform the Integration

After performing the integration, we find: \[ \int \sqrt{1 + 2x^4} \cdot (8x^3) \, dx = \frac{4}{3} x^4 \sqrt{1 + 2x^4} + \frac{2}{3} \sqrt{1 + 2x^4} + C \] where \(C\) is the constant of integration.