After having sufficient experience with differential calculus and its rules, most calculus courses transition into integral calculus. For me, all these new integration rules were hard to wrap my mind around; but, I eventually got them down pat just by studying them and doing problems involving them. I, like my teacher, believe the best way to become an expert at solving calculus integrals is by doing them.
Thankfully, there are fewer rules you must memorize for integration, simply because integration allows for the use of various tricks and tactics to solve problems. I have an article for the most common calculus integration technique: u-substitution. Also, for integrals that cannot be solved with u-substitution, there is integration by parts, which you can also read about.
This article will not explain in-depth what the integral is; it is understood you have a basic understanding that the integral is calculated with antidifferentiation. This calculus tutorial on integration rules may reference differentiation rules, so I certainly recommend that you read my article on how to derive functions, as well.
Although the formulas in this article are the ones used for solving definite integrals (the area under the curve), the focus of this article will be on solving for the indefinite integral (the infinite number of antiderivatives a function has).
It is essential to discuss the added C (the constant of integration) that is used when we solve for an indefinite integral. The indefinite integral of a function is, simply, its antiderivative. However, differentiation rules state that the derivative of a constant is 0. Every real function you will antiderive is added to 0, and thus, you must accommodate for that by adding the constant C at the end.
Graphically speaking, adding a constant to a function only raises or lowers it. It does not change the rate at which the function goes along, or its derivative, at any point. Thus, all constants must be accounted for when solving for an indefinite integral.
The constant of integration can only be solved for if given an initial condition; that is, a point on the indefinite integral function is known.
Integrating a constant is nothing more than slapping the variable of integration after it and adding C. For instance, the indefinite integral of 9 is 9x + C. This makes good sense and can be easily seen as the reverse of the constant rule of differentiation.
Sum and difference rules
Much like the case is with differentiation, integrating a sum or difference is as easy as taking each of the individual addends and adding the results together. The formula:
Just like differentiation, it is crucial that you do not try to separate products or quotients. Unlike differentiation, there isn't a product or quotient rule for integration; rather, these must be handled with tactics such as u-substitution.
Things get a little nastier here, but not by much. It should be noted that the power rule only works if the exponent is a constant, not a variable. Variable exponents require use of the exponential rule.
The image below depicts the power rule for integration:
where n is a constant. In English, you add one to the power and divide the expression by the new power. For example, let us integrate x4.
As you can see, we add 1 to 4 to get 5, and then divide by that 5 and add C to get the answer.
Taking out constants
I will also mention another special thing about constants and integration: they can be put aside until you've finished integrating. Let us say we wanted to integrate 7x. To do this, we would pull the 7 out so we have:
We would then solve the integral using the power rule, and then just multiply by the 7:
Now let's look to see how to solve integrals with variable exponents:
where a is a constant.
So, if you're dealing with a constant being raised to x, just take that entire term and divide it by the natural logarithm of the constant.
Like the case is with derivatives, constant e in indefinite integrals makes things simpler. Since the natural log of constant e is 1, the indefinite integral of ex is just ex + C. Neat!
If your exponent is something other than x, or if you are powering anything but a constant, I recommend using u-substitution to reduce error as much as possible. The fewer integration rules you have to memorize, the better.
Absolute value for logarithms
In any case where your antiderivative leads you to a logarithm, it is important to remember that you cannot log a negative number. Thus, you must use the absolute value. For instance, say I want to find the indefinite integral of 1/x. Right away, I recognize 1/x as the derivative of ln x; however, we must accommodate for the negative xs, and thus, the true indefinite integral is ln |x|.
You'll notice the list for integration rules is quite shorter than that for differentiation rules. This is one thing that makes integral calculus easier to get a handle on. If you come across calculus integrals that cannot be solved with the above rules alone (as will most likely be the case), look into u-substitution and integration by parts. Once you are comfortable with integration as a whole, learn how to find volumes of figures with calculus formulas. Good luck with your studies!