Problem 9 - Essential Calculus

Question

Find the indefinite integral and check the result by differentiation. $$ \int x^{3}\left(x^{4}+3\right)^{2} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(x^3(x^4 + 3)^2 dx\) is \(\frac{(x^4 + 3)^3}{12} + C\), where \(C\) is the constant of integration.

Step 1: Identification

Identify that the given integral \(\int x^3(x^4 + 3)^2 dx\) can be solved using the substitution method. Let \(u = x^4 + 3\). Also, the differential of \(u\) is \(du = 4x^3 dx\), which simplifies the problem substantially.

Step 2: Substitution

Substitute \(u\) and \(du\) into the integral and simplify. Recalling that we found \(du = 4x^3 dx\), we have to compensate for the extra 4. Rewriting the integral we get \(\frac{1}{4} \int u ^2 du\). Note the \(1/4\) comes in because when we divide \(4x^3 dx\) by 4 it becomes \(x^3 dx\).

Step 3: Finding the Integral

Apply the power rule, which states that the integral of \(u ^n\) is \(\frac{u^{n+1}}{n+1}\). For the integral, this becomes \(\frac{1}{4} * \frac{u^{3}}{3} = \frac{u^3}{12}\).

Step 4: Resubstitution

Substitute back the original \(u = x^4 + 3\) into \(\frac{u^{3}}{12}\) to get \(\frac{(x^4 + 3)^3}{12}\). This is the indefinite integral.

Step 5: Differentiation

Differentiate \(\frac{(x^4 + 3)^3}{12}\) to check the result. Using the chain rule and power rule, the derivative is found to be \(x^3(x^4 + 3)^2\), which matches with the original integrand thus validating the solution.