Question 33.1: Two Similar Integrals Calculate the integrals: 1. ∫4/x²+9 dx......
Two Similar Integrals
Calculate the integrals:
\mathbf{1}\!\cdot\int\!{\frac{\ 4}{x^{2}+9}}d x\,.
\mathbf{2}\!\cdot\int\!{\frac{\ 4}{x^{2}+9}}d x\,.
1. For this integral you use the arctangent rule:
\int\!{\frac{4}{x^{2}+9}}d x\ =\ 4\int{\frac{1}{x^{2}+3^{2}}}d x\ =\ 4\Bigg({\frac{1}{3}}\arctan{\frac{x}{3}}\Big)+C\ =\ {\frac{4}{3}}\arctan{\frac{x}{3}}+C\ .
2. This time you should use integration by substitution, with u = x² , du = 2x dx :
\int\!{\frac{4x}{x^{2}+9}}d x~=~2 \int\!{\frac{1}{x^{2}+9}}\ \!\left(2x\right)\!d x~=~2\ln\left\vert x^{2}+9\right\vert+C~=~2\ln\left(x^{2}+9\right)\!+C~=~\ln\!\left(x^{2}+9\right)\!^{2}+C~.
Notice the use of logarithm properties to simplify the answer for the second integral.
Integration problems can require cleverness and creativity. The following example illustrates a standard technique: multiplication and division by a special factor.
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