## Final Answer

## Step-by-step explanation

Problem to solve:

Choose the solving method

We can solve the integral $\int x\cos\left(2x^2+3\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x^2+3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

Isolate $dx$ in the previous equation

Substituting $u$ and $dx$ in the integral and simplify

Take the constant $\frac{1}{4}$ out of the integral

Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$

Replace $u$ with the value that we assigned to it in the beginning: $2x^2+3$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$