Problem 18 - Essential Calculus

Question

Find the indefinite integral and check the result by differentiation. $$ \int(1+3 t) t^{2} d t $$

Step-by-Step Solution

Answer

The indefinite integral of \(x^{2}+\frac{1}{(3 x)^{2}}\) is \(\frac{x^{3}}{3} -\frac{1}{9x} + C\).

Step 1: Identify the elements to integrate individually

The function \(x^{2}+\frac{1}{(3 x)^{2}}\) can be broken down into two parts: \(x^{2}\) and \(\frac{1}{(3 x)^{2}}\). Both of these parts should be handled separately.

Step 2: Apply the power rule to \(x^{2}\)

The power rule states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\). This gives us the integral of \(\int x^{2} dx = \frac{x^{3}}{3}\).

Step 3: Transform \(\frac{1}{(3x)^{2}}\) into a suitable form

Rewrite \(\frac{1}{(3x)^{2}}\) as \(\frac{1}{9} * \frac{1}{x^{2}} = \frac{1}{9x^{2}}\). The constant \(\frac{1}{9}\) can be moved in front of the integral.

Step 4: Apply the power rule to \(\frac{1}{9x^{2}}\)

Applying the power rule again to \(\frac{1}{9x^{2}}\) gives \(\int \frac{1}{9x^{2}} dx = -\frac{1}{9x}\).

Step 5: Combine results

Adding these two results gives the indefinite integral: \(\frac{x^{3}}{3} -\frac{1}{9x} + C\), where \(C\) is the constant of integration.

Step 6: Check the result by differentiation

The derivative of \(\frac{x^{3}}{3} -\frac{1}{9x} + C\) is \(x^{2} + \frac{1}{(3 x)^{2}}\), which confirms that the integral was calculated correctly.