Integration by parts is a unique fashion of integrating two functions when they’re multiplied together or for evaluating integrals whose integrand is the product of two functions to make complicated integrations easy to break. This system is also nominated as partial integration. This is one of the exclusive topics tested in AP Calculus BC.
Formula : Integration by Parts
The following equation gives the formula for integrating by parts;
![\[\int f(x) g(x) d x=f(x) \int g(x) d x-\int\left[f^{\prime}(x) \int g(x) d x\right] d x\]](/wp-content/ql-cache/quicklatex.com-903f44c059f9c3ee9ae2b66b64cc8108_l3.png "Rendered by QuickLaTeX.com")
ILATE Rule
In integration by parts, we’ve learnt when the product of two functions is given to us further, we apply the needed formula. The integral of the two functions is taken by considering the left term as the first function and the other term as the alternate function. This system is called the Ilate rule. Suppose we’ve to integrate
![\[x \cos x\]](/wp-content/ql-cache/quicklatex.com-646e565b9bd49b72154422e84ac193cc_l3.png "Rendered by QuickLaTeX.com")
; in this case, we consider x as the first function and cos x as the alternate function. So mainly, the first function is chosen so that the derivative of the function could be fluently integrated. Generally, the preference order of this rule is grounded on some functions similar as Inverse, Algebraic, Logarithm, Trigonometric, and Exponent.
To summarise the way of applying ILATE rule we can follow the below-given steps,
- Identify the type of each function as Inverse trigonometric( I), Logarithmic( L), Algebraic( I), Trigonometric( T), or Exponential( E).
- Now precisely choose which of the given functions comes first in the order of ILATE( or) LIATE and let it be as the first function.
- Elect the remaining function as the alternate function.
- Then apply the integration by parts formula.
For example, Question : Integrate
![\[\int x \ln x d x\]](/wp-content/ql-cache/quicklatex.com-5377634603343e91971dcc2737d62b7b_l3.png "Rendered by QuickLaTeX.com")
Solution:
Now applying the above-given steps,
- x is the algebraic function and ln x is the logarithmic function.
- Among Algebraic (A) and Logarithmic (L) functions, we know that the Logarithmic (L) function comes first in ILATE rule. So the first function is ln x as per ILATE rule.
- The remaining function, x, is elected as the second function.
By applying the integration by parts formula, we get the following:
![\[\begin{aligned}& \int x \ln x d x=(\ln x) \int x d x-\int\left[d / d x(\ln x) \int x d x\right] d x \\& =(\ln x)\left(x^2 / 2\right)-\int(1 / x)\left(x^2 / 2\right) d x \\& =\left(x^2 \ln x\right) / 2-(1 / 2) \int x d x \\& =\left(x^2 \ln x\right) / 2-(1 / 2)\left(x^2 / 2\right) d x \\& =\left(x^2 \ln x\right) / 2-\left(x^2 / 4\right)+C\end{aligned}\]](/wp-content/ql-cache/quicklatex.com-7e6d50330c231e7111dba9592e613664_l3.png "Rendered by QuickLaTeX.com")
Important points regarding Integration by parts:
- Integration by parts isn’t applicable for functions similar to
![\[\int \sqrt{x} \cos x d x\]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDUiIGhlaWdodD0iNDAiIHZpZXdCb3g9IjAgMCAxMDUgNDAiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- We don’t add any constant while finding the integral of the alternate function.
- Generally, if any function is a power of x or a polynomial in x, we take it as the first function. Even so, in cases where another function is an inverse trigonometric or logarithmic function, we take them as the first function.
- Apart from the Integration by parts method, we use two other methods which are used to perform the integration. They are:
![\[\int \sqrt{x} \cos x d x\]](/wp-content/ql-cache/quicklatex.com-9205100f9c9294286ff8495af18a635e_l3.png "Rendered by QuickLaTeX.com")
- Integration by Substitution
- Integration using Partial Fractions
While Substitution is a part of AP Calculus AB and BC both, Integration by parts and Partial fractions are tested only in AP Calculus BC.
To know more about the other chapters tested in Calculus BC, Check out Everything you need to know about AP Calculus BC!
