Integration by Parts

Integration by Parts

Integration by parts is a technique used in calculus to find the integral of a product of functions. It is particularly useful when dealing with complex functions that cannot be integrated by other techniques, such as substitution or partial fractions. The method is based on the product rule of differentiation, which states that:

$(uv)' = u'v + uv'$

where $u$ and $v$ are functions of $x$, and $'$ denotes differentiation with respect to $x$.

Using the product rule, we can derive the integration by parts formula:

$\int u\,dv = uv - \int v\,du$

where $u$ and $v$ are functions of $x$, and $du$ and $dv$ denote their respective differentials.

To use the integration by parts formula, we need to choose two functions $u$ and $dv$ such that the integral on the right-hand side is easier to evaluate than the original integral. We then differentiate $u$ and integrate $dv$ to obtain $du$ and $v$, respectively, and substitute these into the formula to find the value of the original integral.

Let's consider an example to illustrate the integration by parts method. Suppose we want to find the integral of:

$\int x^2 e^x \,dx$

We can choose $u = x^2$ and $dv = e^x\,dx$, so that:

$du = 2x\,dx \quad \textrm{and} \quad v = e^x$

Substituting these into the integration by parts formula, we get:

$\int x^2 e^x\,dx = x^2 e^x - \int 2xe^x\,dx$

We can now use integration by parts again to evaluate the integral on the right-hand side. Choosing $u = 2x$ and $dv = e^x\,dx$, we have:

$du = 2\,dx \quad \textrm{and} \quad v = e^x$

Substituting these into the formula, we get:

$\int 2xe^x\,dx = 2xe^x - \int 2e^x\,dx = 2xe^x - 2e^x + C$

where $C$ is the constant of integration. Substituting this result back into the original formula, we obtain:

$\int x^2 e^x\,dx = x^2 e^x - 2xe^x + 2e^x + C$

This is the final answer.

Integration by parts is a powerful technique that allows us to evaluate a wide range of integrals. It is particularly useful in applications such as physics and engineering, where integrals often arise in the context of solving differential equations. By choosing the right functions $u$ and $dv$, we can simplify the integral and make it easier to evaluate. With practice, integration by parts can become a valuable tool in the calculus toolbox.

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