Final Answer
Step-by-step Solution
Problem to solve:
Specify 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
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 $f'(x) = nx^{n-1}$
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
Simplify the fraction $\frac{x\cos\left(u\right)}{4x}$ by $x$
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{4}$ out of the integral
We can solve the integral $\int\cos\left(u\right)du$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
The integral of a constant is equal to the constant times the integral's variable
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Any expression multiplied by $1$ is equal to itself
Now replace the values of $u$, $du$ and $v$ in the last formula
Multiply the single term $\frac{1}{4}$ by each term of the polynomial $\left(u\cos\left(u\right)+\int u\sin\left(u\right)du\right)$
We can solve the integral $\int u\sin\left(u\right)du$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
The derivative of the linear function is equal to $1$
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Any expression multiplied by $1$ is equal to itself
Now replace the values of $u$, $du$ and $v$ in the last formula
Multiplying the fraction by $-1$
Multiply the single term $\frac{1}{4}$ by each term of the polynomial $\left(-u\cos\left(u\right)+\int\cos\left(u\right)du\right)$
Divide $1$ by $4$
Multiplying the fraction by $u\cos\left(u\right)$
Divide $1$ by $4$
Combining like terms $\frac{1}{4}u\cos\left(u\right)$ and $\frac{-u\cos\left(u\right)}{4}$
Divide $-1$ by $4$
Subtract the values $\frac{1}{4}$ and $-\frac{1}{4}$
Any expression multiplied by $0$ is equal to $0$
$x+0=x$, where $x$ is any expression
Simplify the expression inside 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 $C$