## Final Answer

## Step-by-step Solution

Problem to solve:

Specify the solving method

First, factor the terms inside the radical by $2$ for an easier handling

Learn how to solve integrals with radicals problems step by step online.

$\int t\sqrt{2\left(t^2+\frac{3}{2}\right)}dt$

Learn how to solve integrals with radicals problems step by step online. Find the integral int(t(2t^2+3)^1/2)dt. First, factor the terms inside the radical by 2 for an easier handling. Taking the constant out of the radical. We can solve the integral \int\sqrt{2}t\sqrt{t^2+\frac{3}{2}}dt by applying integration method of trigonometric substitution using the substitution. Now, in order to rewrite d\theta in terms of dt, we need to find the derivative of t. We need to calculate dt, we can do that by deriving the equation above.