Question

how to know if should use the u-substitution or integrating by parts in integrating a function?

Answer 1

there are some standard expressions with standard substitutions, like
\(\begin{array}{|c|c|}\hline \text{Expression in Integral} &Substitution
\\ \hline \sqrt{a^2-x^2}&x=a\sin \theta \quad or \quad x=a\cos\theta
\\ \hline \sqrt{x^2-a^2}&x=a\sec \theta \quad or \quad x=a\csc\theta
\\ \hline x^2+a^2 &x=a\tan \theta \quad or \quad x=a\cot\theta
\\ \hline \sqrt{\frac{a-x}{a+x}}& x=a\cos2\theta
\\ \hline \sqrt{\frac{a-x}{x}}or\sqrt{\frac{x}{a-x}} & x=a\sin^2\theta
\\ \hline \sqrt{\frac{a+x}{x}}or\sqrt{\frac{x}{a+x}} & x=a\tan^2\theta
\\ \hline \sqrt{2ax+x^2} & x=2a\tan^2\theta
\\ \hline \sqrt{2ax-x^2} & x=2a\sin^2\theta
\\ \hline \sqrt{\frac{a^2-x^2}{a^2+x^2}} & x^2=a^2\cos2\theta


\\ \hline \end{array}\)
\(\begin{array}{|c|c|}\hline \text{Expression in Integral} &Substitution
\\ \hline \ln|f(x)| & u=ln|f(x)|
\\ \hline \ln|f(x)|\pm \ln|g(x)| & u=ln|f(x)| )|\pm \ln|g(x)|
\\ \hline f(x)^nf’(x) & u=f(x)
\\ \hline e^{f(x)} \quad or \quad a^{f(x)} & u=f(x)

\\ \hline \sqrt{ax+b} \\ \frac{cx+d}{\sqrt{ax+b} }\\(cx+d) \sqrt{ax+b} & u= \sqrt{ax+b}
\\ \hline \frac{\sin \:x+\cos \:x}{a+b\sin\:2x} & u=\int Numerator
\\ \hline P(x)(ax+b)^n \\ \text{P(x)is any polynomial in x} & u=ax+b
\\ \hline \frac{1}{x^{1/m}+x^(1/n)} & x=t^k,k=LCM(m,n)



\\ \hline \end{array}\)
you can use these or try to bring your integral in any of these form by manipulation.
For integration by parts, you really need 2 different functions, like a logaritmic and trigonometric product , or algebraic and ecponential product...then integration by parts is best option.

Answer 2

do you have any specific example in which you have difficulty integrating ??