Apply the trigonometric identity: $\cos\left(\theta \right)^2$$=\frac{1+\cos\left(2\theta \right)}{2}$
$\int x\frac{1+\cos\left(2x\right)}{2}dx$
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We can solve the integral $\int x\frac{1+\cos\left(2x\right)}{2}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
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
$u=2x$
Intermediate steps
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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
$du=2dx$
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Isolate $dx$ in the previous equation
$du=2dx$
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Substituting $u$ and $dx$ in the integral and simplify
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.