We can solve the integral $\int x\left(1+\cos\left(2x\right)\right)dx$ 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
$u=2x$
Intermediate steps
5
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
$du=2dx$
6
Isolate $dx$ in the previous equation
$du=2dx$
Intermediate steps
7
Rewriting $x$ in terms of $u$
$x=\frac{u}{2}$
Intermediate steps
8
Substituting $u$, $dx$ and $x$ in the integral and simplify
We can solve the integral $\int u\left(1+\cos\left(u\right)\right)du$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more
Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.