Integration by Parts of Indefinite Integrals Examples 2
We will now look at some more examples of Integration by Parts of Indefinite Integrals. Please check our the Integration by Parts Examples 1 first before reviewing these examples.
(1)Example 1
Evaluate the integral $\int \ln x \: dx$.
At first this example may not look that suitable for integration by parts, however, if we let $u = \ln x$, while when differentiated is simpler, and let $dv = dx$, then we can easily apply integration by parts. Therefore $du = \frac{1}{x} \: dx$ and $v = x$. Substituting this into our integration by parts formula we get:
(2)Example 2
Evaluate the integral $\int e^x \cos x \: dx$.
We note that choosing either $u = e^x$ or $u = \cos x$ for differentiation with both result in functions that are relatively the same in terms of difficulty. Let's choose $u = e^x$ though, and therefore $dv = \cos x \: dx$. Then $du = e^x \: dx$ and $v = \sin x \: dx$. Substituting this into the integration by parts formula we obtain that:
(3)Now we will apply integration by parts again. Let $u = e^x$ and $dv = \sin x \: dx$. Therefore, $du = e^x \: dx$ and $v = -\cos x$. Substituting this back in we get:
(4)Example 3
Evaluate the integral $\int \sin ^{-1} (3x) \: dx$.
We will choose $u = \sin ^{-1} (3x)$ since when differentiated, becomes simpler. Therefore, $dv = dx$. Therefore, $du = \frac{3}{\sqrt{1 - (3x)^2}}$ and let $v = x$. Making the appropriate substitutions we get that:
(5)We will now use the technique of U-Substitution of Indefinite Integrals. Let $u = 1 - 9x^2$ so that $du = -18x \: dx$ and $\frac{-1}{6} du = 3x \: dx$. Therefore:
(6)Example 4
Evaluate the integral $\int 2x \tan ^{-1} x \: dx$.
We will choose $u = \tan^{-1} x$ since when differentiated it becomes simpler. Therefore $dv = 2x \: dx$. Thus, $du = \frac{1}{1 + x^2}$ and $v = x^2$. Substituting into the integration by parts formula we obtain that:
(7)Now note that $\frac{x^2}{1 + x^2} = \frac{x^2 + 1 - 1}{1 + x^2} = \frac{(x^2 + 1) - 1}{(x^2 + 1)} = 1 - \frac{1}{1 + x^2}$. Therefore:
(8)