Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Integrate by parts
- Integrate by partial fractions
- Integrate by substitution
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL 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
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
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
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
The integral of a constant is equal to the constant times the integral's variable
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
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
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 $\left(-u\cos\left(u\right)+\int\cos\left(u\right)du\right)$
Simplify the expression
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 $C$