$$\int {x^2\over x^3+2}dx$$
I know I can use u-sub with $u=x^3+2$ and $du=3x^2dx$ to get:
$${\ln(x^3+2)\over3}$$
But what about a problem like:
$$\int 35\sqrt x e^\sqrt x dx$$
I first tried integrating by parts because I thought of it as functions $35\sqrt x$ and $e^\sqrt x$ multiplied together. I didn't know to set $u=\sqrt x$ until I got ${70\over 3}x^{3\over2}-35\int xe^\sqrt x$ through integration by parts (and then looked it up on an online calculator), and I'm assuming that would go into another loop of integration by parts because of the way derivatives with $e$ works.
How do I know whether to try a u-substitution or integration by parts first when there isn't an apparent function I can differentiate and easily replace within the integral? This includes problems with and without $e$.
Best Answer
In $\int \frac{x^2}{x^3+2}$ you need to take the $u$ substitution of a term that cancels the numerator. Here if we take $u=x^3+2$ then $dx=\frac{du}{3x^2}$. So the $x^2$ term in the numerator gets canceled.
In the cases like $\int(6x^2)(2x^3+5)^6dx$ it is always advisable to take the number with the highest power. In this case $u=2x^3+5$
In the cases like $\int cos(5x-7)dx$ it is always advisable to take the number in the brackets. In this case $u=5x-7$
In the cases like $\int 35\sqrt{x}e^{\sqrt{x}}$ it is always advisable to take the exponent of $e$ as $u$. In this case $u=\sqrt{x}$
Another example of this is $\int \frac{e^x}{e^x}dx=\int1+e^{-x}dx$, even in this case we take the exponent of e as $u$.
$u=-x$