Here, we show you a step-by-step solved example of calculus. This solution was automatically generated by our smart calculator:
We can solve the integral $\int x\sin\left(2x\right)dx$ 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
We can solve the integral $\int\sin\left(2x\right)dx$ 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 $2x$ 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 $u=2x$
Find the derivative
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
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
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
The integral of a function times a constant ($-\frac{1}{2}$) is equal to the constant times the integral of the function
Now replace the values of $u$, $du$ and $v$ in the last formula
Apply the formula: $\int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)+C$, where $a=2$
Simplify the expression inside the integral
The integral $\frac{1}{2}\int\cos\left(2x\right)dx$ results in: $\frac{1}{4}\sin\left(2x\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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