I am trying to learn calculus as part of a catch up course for a degree. Finding the derivative of a function makes sense to me but finding the integral of a function: I just can't get my head around.
This is the question: Find the indefinite integral and evaluate the definite integral from $-1$ to $+1$.
The function is written like this: $$f(x) = 2x^2 - 3x + 5$$ Could someone break down the method for integration of the above into small steps that I can wrap my head around? Thanks!
On
This question can be answered by the basic property of integral which states that an integral with a polynomial function will obey a distributive property. That is,
$$\int(f(x) + g(x))dx =\int f(x)dx + \int g(x)dx$$ so that \begin{align}\int(2x^2 – 3x + 5) dx & =\int 2x^2dx - \int 3x dx + \int5 dx \\ & = \frac{2x^3}{3} – \frac{3x^2}{2} + 5x + c\end{align}
The indefinite integral of $ax^n$ is $\frac{a}{n+1}x^{n+1}$ and the integral of a sum is the sum of the integrals of the summands. Have you tried to sit down and work that out? For the definite integral from $A$ to $B$ you evaluate $F(B)-F(A)$, where $F$ is the primitive function (=the indefinite integral up to constants) of your function $f$.
I will add more details if you let me know where problems are arising for you.