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# Integration techniques

## Integration by parts

Integration by parts applies to the product of two functions.

Integration by parts states that, if $f$ and $g$ are two functions, $$\int_a^b f'(x)g(x)\,dx = -\int_a^b f(x)g'(x)\,dx + \Big[ fg\Big]_a^b.$$ Here, we use the notation $\Big[ f \Big]_a^b = f(b) - f(a)$.

With $f(x) = \ln x$, $g(x) = x$ so that $f'(x) = 1/x$, $g'(x) =1$, $$\int_1^2\ln x\,dx = \Big[x\ln x\Big]_1^2 - \int_1^2\,dx = 2\ln 2-1.$$

LIATE is a common rule to determine which type of function to differentiate first. It stands for Logarithmic ($\ln x$), Inverse trigonometric ($\arcsin x$), Algebraic ($x^n$), Trigonometric ($\sin x$), Exponential ($e^x$). To integrate the product of two functions, the function that comes first in this list is taken as the $g$ in the above formula.

In the example above, LIATE says that we should differentiate $\ln x$ and integrate the algebraic function $1$. A variant is ILATE.

The integration by parts formula is proved simply by noting that the derivative of $fg$ is $f'g+fg'$. By the fundamental theorem of calculus $$\Big[ fg\Big]_a^b = \int_a^b(fg)'\,dx = \int_a^b f'(x)g(x)\,dx + \int_a^b f(x)g'(x)\,dx.$$

## Integration by substitution

Integration by substitution is a useful technique to compute integrals when the integrand depends on the composition of two functions.

Integration by substitution states that, if $f$ and $g$ are two functions, $$\displaystyle \int_{f(a)}^{f(b)}g(y)\,dy = \int_a^b f'(x) g\big(f(x)\big) \,dx$$

Setting $f(x) = x^2$ and $g(y) = e^y$, we obtain $$\int_0^2 2x e^{x^2}\,dx = \int_0^4 e^{y}\,dy = \Big[e^y\Big]_0^4 = e^4 - 1$$

Informally, integration by substitution corresponds to the change of variable $y= f(x)$. We have that $dy = f'(x)\,dx$. Moreover, we see that when $y$ varies between the limits $f(a)$ and $f(b)$, $x$ varies between $a$ and $b$.

Rigorously, the integration by parts formula is an application of the chain rule, i.e. the differentiation rule for the composition of functions.

Proof If $G$ is a primitive of $g$, the derivative of $G\circ f$ is $f'\, G'\circ f = f'\,g\circ f$. So the fundamental theorem of calculus reads \begin{gather*} \int_{f(a)}^{f(b)}g(y)\,dy = G\circ f(b) - G\circ f(a) \\ = \int_a^b (G\circ f)'\,dx = \int_a^b f'(x)g\big(f(x)\big)\,dx \end{gather*}