The integration by part
The integration by part is a consequent of the multiplication rule of derivative and the fundamental theorem of Calculus. Using $d(uv) = udv + vdu$, we derive \(\int udv = uv-\int vdu.\)
The followings are examples:
Example 1 $\int \ln(x) dx$
Example 2 $\int x\sin(x) dx$
Example 3 $\int x^2e^x dx$
Example 4 $\int e^x\sin(x) dx$
Example 5 $\int tan^{-1}(x)dx$ and $\int\sin^{-1}(x)dx$.
Example 6 $\int \sin^2(x)dx$
Trigonometric integrals $\int \sin^{m}(x)\cos^n(x)dx$
To integrate $\sin^{m}(x)\cos^{n}(x)$, we consider the following cases:
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If $m$ or $n$ is odd, say $m$ is odd, then we will use the substitution rule by letting $u=\cos(x)$. Using the trig identity $\sin^2(x) = 1-\cos^2(x)$, the integrals of trig functions become integrals of polynomials.
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If both $m$ and $n$ are even, we can use integration by part to reduce the $m$ and $n$ by $2$, i.e. an integral of $\sin^m(x)\cos^n(x)$ becomes an integral of $\sin^{m-2}(x)\cos^n(x)$ or $\sin^{m}(x)\cos^{n-2}(x)$ depending on how you set up your $u$ and $dv$. Another simpler method is use the double angle formulas: \(\sin^2(\theta) = \frac{1-\cos(2\theta)}{2}\text{ and }\cos^{2}(\theta)=\frac{1+\cos(2\theta)}{2}.\)
| m\n | $n$ is odd | $n$ is even |
|---|---|---|
| $m$ is odd | use 1. by setting either $u=\cos(x)$ or $u=\sin(x)$ | use 1. by setting $u=\cos(x)$ |
| $m$ is even | use 1. by setting $u=\sin(x)$ | Use double angle formula or integration by part to reduce the degree $m$ and $n$ until do able |