Here, we show you a step-by-step solved example of cyclic integration by parts. This solution was automatically generated by our smart calculator:
We can solve the integral $\int e^{-x}\cos\left(2x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
The derivative of the 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 $f'(x) = -\sin(x)\cdot D_x(x)$
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
We can solve the integral $\int e^{-x}dx$ 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 $-x$ 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 $u=-x$
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
Move up the $-1$ from the denominator
Substituting $u$ and $dx$ in the integral and simplify
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$
Replace $u$ with the value that we assigned to it in the beginning: $-x$
Replace $u$ with the value that we assigned to it in the beginning: $-x$
Multiply $-2$ times $-1$
The integral of a function times a constant ($2$) is equal to the constant times the integral of the function
Now replace the values of $u$, $du$ and $v$ in the last formula
We can solve the integral $\int e^{-x}\sin\left(2x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
We can solve the integral $\int e^{-x}dx$ 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 $-x$ 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
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$
Replace $u$ with the value that we assigned to it in the beginning: $-x$
Now replace the values of $u$, $du$ and $v$ in the last formula
Multiply the single term $-2$ by each term of the polynomial $\left(-e^{-x}\sin\left(2x\right)+2\int e^{-x}\cos\left(2x\right)dx\right)$
The integral $-2\int e^{-x}\sin\left(2x\right)dx$ results in: $2e^{-x}\sin\left(2x\right)-4\int e^{-x}\cos\left(2x\right)dx$
This integral by parts turned out to be a cyclic one (the integral that we are calculating appeared again in the right side of the equation). We can pass it to the left side of the equation with opposite sign
Moving the cyclic integral to the left side of the equation
Adding the integrals
Move the constant term $5$ dividing to the other side of the equation
The integral results in
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 $C$
Multiply the single term $\frac{1}{5}$ by each term of the polynomial $\left(-e^{-x}\cos\left(2x\right)+2e^{-x}\sin\left(2x\right)\right)$
Multiplying the fraction by $-1$
Expand and simplify
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