| Version 4 |
Version 3 |
| When one has an integral of a product of two functions, it is usually preferable to simplify the integrand by integrating one of the functions and differentiating the other. This process is called integrating by parts, and is defined in the following way, where $u$ and $v$ are functions of $x$. |
When one has an integral of a product of two functions, it is usually preferable to simplify the integrand by integrating one of the functions and differentiating the other. This process is called integrating by parts, and is defined in the following way, where $u$ and $v$ are functions of $x$. |
| $$\int u\cdot v'\; dx = u\cdot v - \int v\cdot u'\; dx$$ |
$$\int u\cdot v'\; dx = u\cdot v - \int v\cdot u'\; dx$$ |
| This process may be repeated indefinitely, and in some cases it may be used to solve for the original integral algebraically. For definite integrals, the rule appears as |
This process may be repeated indefinitely, and in some cases it may be used to solve for the original integral algebraically. For definite integrals, the rule appears as |
| $$\int_a^b u(x)\cdot v'(x)\; dx = (u(b)\cdot v(b)-u(a)\cdot v(a)) - \int_a^b v(x)\cdot u'(x)\; dx$$ |
$$\int_a^b u(x)\cdot v'(x)\; dx = (u(b)\cdot v(b)-u(a)\cdot v(a)) - \int_a^b v(x)\cdot u'(x)\; dx$$ |
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\textbf{Proof:}
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\textbf{Proof :}
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| Integration by parts is simply the antiderivative of a product rule. Let $G(x)=u(x)\cdot v(x)$. Then, |
Integration by parts is simply the antiderivative of a product rule. Let $G(x)=u(x)\cdot v(x)$. Then, |
| $$G'(x) = u'(x)v(x) + u(x)v'(x)$$ |
$$G'(x) = u'(x)v(x) + u(x)v'(x)$$ |
| Therefore, |
Therefore, |
| $$G'(x) - v(x)u'(x) = u(x)v'(x)$$ |
$$G'(x) - v(x)u'(x) = u(x)v'(x)$$ |
| We can now integrate both sides with respect to $x$ to get |
We can now integrate both sides with respect to $x$ to get |
| $$G(x) - \int v(x)u'(x)\; dx = \int u(x)v'(x)\; dx$$ |
$$G(x) - \int v(x)u'(x)\; dx = \int u(x)v'(x)\; dx$$ |
| which is just integration by parts rearranged. \\ |
which is just integration by parts rearranged. |
| \textbf{Example:} We integrate the function $f(x)=x\sin x$: Therefore we define $u(x):=x$ and $v'(x)=\sin x$. So integration by parts yields us: |
[nice abstract pic and examples coming] |
| $$\int x\sin x\mathit{dx}=-x\cos x+\int\cos x\mathit{dx}=-x\cos x+\sin x.$$ |
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