** Final Answer

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** Step-by-step Solution **

** Specify the solving method

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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

Differentiate both sides of the equation $u=2x^2+3$

Find the derivative

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($3$) is equal to zero

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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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

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Isolate $dx$ in the previous equation

Simplify the fraction $\frac{x\cos\left(u\right)}{4x}$ by $x$

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

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

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Divide $1$ by $4$

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

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

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

** Final Answer

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