Question

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(2 t^{2}-1\right)^{2} d t $$

Short answer

The indefinite integral of \((2t^2 - 1)^2\) is \(\frac{4}{5}t^5 - \frac{4}{3}t^3 + t + C\).

Step-by-step solution

Expand the integrand

We need to expand the term \((2t^2 - 1)^2\) first. We do this by applying the formula \((a - b)^2 = a^2 - 2ab + b^2\). So, it becomes \(4t^4 - 4t^2 + 1\). Now, we can rewrite the integral as \(\int 4t^4 dt - \int 4t^2 dt + \int dt\).

Apply the power rule

Now we can apply the power rule for each term. For the first term we get \(\frac{4}{5}t^5\), for the second term we get \(-\frac{4}{3}t^3\), and integrating the last term gives \(t\).

Combine Results and Add Constant of Integration

The indefinite integral is found by adding up the results of each term and including the constant of integration (C). So, the solution to the indefinite integral is given by: \(\frac{4}{5}t^5 - \frac{4}{3}t^3 + t + C\).