Exams · UNT (Kazakhstan), mathematics
Antiderivatives and integrals
Find an antiderivative and compute a definite integral.
How to solve it
Antiderivatives
An antiderivative is a function $F$ whose derivative is the given one: $F' = f$. Finding it means reading the table of derivatives backwards. The answer can always be checked: differentiate it and compare with what was under the integral.
Step by step
- The table: $\int x^n\,dx = \frac{x^{n + 1}}{n + 1} + C$ for $n \ne -1$, $\int \frac{dx}{x} = \ln|x| + C$, $\int \cos x\,dx = \sin x + C$, $\int e^x\,dx = e^x + C$.
- When the function is applied to $kx + b$, divide the result by $k$: $\int \cos 3x\,dx = \frac{\sin 3x}{3} + C$.
- When the derivative of an inner function stands next to it, substitute $t = g(x)$: $\int 2x e^{x^2}\,dx = e^{x^2} + C$.
- A polynomial times $e^x$, $\sin x$ or $\cos x$ is integrated by parts.
- Check the answer by differentiating it.
Common mistakes
- Dividing by $n$ instead of $n + 1$: $\int x^3\,dx = \frac{x^4}{4} + C$.
- Forgetting to divide by $k$ after a linear substitution.
- Not checking the answer, although the check by differentiation takes a minute.
Example
Definite integrals
The definite integral $\int_a^b f(x)\,dx$ is the signed area under the graph. It is computed without any areas at all: find an antiderivative and subtract its values at the ends of the segment.
Step by step
- Find an antiderivative $F(x)$, as in the antiderivative tasks. The constant $C$ is not needed: it cancels in the subtraction.
- Substitute the upper limit, then the lower one, and subtract: $F(b) - F(a)$.
- Subtract $F(a)$ as a whole, in brackets: it may have minuses of its own.
Common mistakes
- Subtracting the other way round: $F(a) - F(b)$ has the opposite sign.
- Losing a minus when $F(a)$ is negative: $F(b) - (-3) = F(b) + 3$.
- Making a mistake in the antiderivative itself. Check it by differentiating before substituting the limits.