It depends whether you mean the indefinite integral (also known as the antiderivative), or the definite integral. In initial calculus courses, you usually start with the indefinite integral.In any case, there is no quick way to explain this; several chapters of calculus books are dedicated to learning several different methods to solve integrals, and those methods don't work in all cases.
In general, you need to go through a calculus course, or book, and learn those methods.
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There are many techniques, integration by parts being one of the ones most commonly used. To get a good answer you would need to ask about how to integrate a specific function.
There are very many ways to solve integrals: it is not a topic which can be covered in a QA forum like this. Even one textbook will not be enough.
By using the fundamental theorem of Calculus. i.e. The integral of f(x) = F(x), your limits are [a,b]. Solve: F(b) - F(a). The FTC, second part, says that if f is a continuous real valued function of [a,b] then the integral from a to b of f(x)= F(b) - F(a) where F is any antiderivative of f, that is, a function such that F'(x) = f(x). Example: Evaluate the integral form -2 to 3 of x^2. The integral form -2 to 3 of x^2 = F(-2) - F(3) = -2^3/3 - 3^3/3 = -8/3 - 27/3 = -35/3
integral (a^x) dx = (a^x) / ln(a)
In order to evaluate a definite integral first find the indefinite integral. Then subtract the integral evaluated at the bottom number (usually the left endpoint) from the integral evaluated at the top number (usually the right endpoint). For example, if I wanted the integral of x from 1 to 2 (written with 1 on the bottom and 2 on the top) I would first evaluate the integral: the integral of x is (x^2)/2 Then I would subtract the integral evaluated at 1 from the integral evaluated at 2: (2^2)/2-(1^2)/2 = 2-1/2 =3/2.
Integral of [1/(sin x cos x) dx] (substitute sin2 x + cos2 x for 1)= Integral of [(sin2 x + cos2 x)/(sin x cos x) dx]= Integral of [sin2 x/(sin x cos x) dx] + Integral of [cos2 x/(sin x cos x) dx]= Integral of (sin x/cos x dx) + Integral of (cos x/sin x dx)= Integral of tan x dx + Integral of cot x dx= ln |sec x| + ln |sin x| + C
The antiderivative, or indefinite integral, of ex, is ex + C.