Question

How do I get this form?

Answer 1

Short answer: you don't.

The linear term in the numerator of the integral means the form shown is not applicable. Rather, you perform the integration using partial fraction expansion.

$\displaystyle\int{\frac{5x+1}{25x^2+60x-13}}\,dx=\int{\frac{5x+1}{(5x-1)(5x+13)}}\,dx\\\\=\frac{1}{35}\int{\frac{5}{5x-1}}\,dx+\frac{6}{35}\int{\frac{5}{5x+13}}\,dx$

The integral is ...

... (1/35)ln|5x-1| +(6/35)ln|5x+13| +C

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If the numerator of your integral were a constant, then the fractions multiplying the separate partial fraction integrals would have the same magnitude and opposite signs. You would end with the difference of logarithms, which could be expressed as the log of a ratio as shown in your problem statement.