Evaluate. (Be sure to check by differentiatingl) \[ \int\left(7+x^{4}\right)^{3} 4 x^{3} d x \] \[ \int\left(7+x^{4}\right)^{3} 4 x^{3} d x=\square \] (Type an exact answer. Use parentheses to clearly denote the argument of each function.)
Solution
Step 1 ：Let's use the substitution method to solve this integral. We notice that the derivative of \(x^4\) is \(4x^3\), which is present in the integral.
Step 2 ：Let \(u = 7 + x^4\), then \(du = 4x^3 dx\). The integral then becomes \(\int u^3 du\), which is straightforward to solve.
Step 3 ：The integral of \(u^3\) with respect to \(x\) is \(\frac{x^{13}}{13} + \frac{7x^9}{3} + \frac{147x^5}{5} + 343x\).
Step 4 ：After substituting \(u = 7 + x^4\) back in, the integral becomes \(\frac{(7 + x^4)^{13}}{13} + \frac{7(7 + x^4)^9}{3} + \frac{147(7 + x^4)^5}{5} + 343(7 + x^4)\). This is the antiderivative of the original function, so it is the solution to the integral.
Step 5 ：Final Answer: \(\boxed{\int\left(7+x^{4}\right)^{3} 4 x^{3} d x = \frac{(7 + x^4)^{13}}{13} + \frac{7(7 + x^4)^9}{3} + \frac{147(7 + x^4)^5}{5} + 343(7 + x^4) + C}\), where \(C\) is the constant of integration.
