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Integration Techniques

Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.

Integration with Partial Fractions

         

If \[f(x)=\int \frac{4x}{x^2-2x-3} dx\] and \(f(0)=15 \ln 3,\) what is the value of \(f(2)?\)

If \(f(x)\) is a function such that \[f'(x)=\frac{1}{25x^2-1}\] and \(f(0)=7,\) what is the value of \(f(1)?\)

What is the indefinite integral \[\int \frac{x+2}{(x-4)(x-5)} dx?\] (Use \(C\) for the constant of integration.)

If \[f(x)=\int \frac{12x^2}{(x-1)(x^2+x+1)} dx\] and \(f(0)=0,\) what is the value of \(\displaystyle{e^{\frac{f(9)}{4}}}?\)

If \(a\) is a nonzero real number, what is \[ \int \frac{35}{x^2 - a^2} dx?\]

Details and assumptions

Use \(C\) as the constant of integration.

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