Antiderivatives and integrals
Antiderivatives from the table, linear substitution and integration by parts, definite integrals by the Newton–Leibniz formula. Every answer can be checked by differentiating.
How to solve it
An antiderivative is a function whose derivative is the given one. So the table of antiderivatives is the table of derivatives read right to left, and every answer can be checked by differentiating. A definite integral is the difference of the antiderivative's values at the ends.
Step by step
- Split the integral of a sum into the integrals of the terms, take constant factors out.
- Powers: $\int x^n\,dx = \frac{x^{n+1}}{n + 1}$ for $n \ne -1$, and $\int \frac{dx}{x} = \ln|x|$.
- With $kx + b$ inside, take the antiderivative as usual and divide by $k$.
- A polynomial times $e^x$, $\sin x$ or $\ln x$: by parts, $\int u\,dv = uv - \int v\,du$.
- A definite integral: find an antiderivative $F$ and compute $F(b) - F(a)$.
Common mistakes
- Forgetting to divide by the coefficient in a linear substitution: $\int \sin 5x\,dx = -\frac{\cos 5x}{5}$.
- Sign errors: the antiderivative of the sine is $-\cos x$, of the cosine $\sin x$.
- Using the power rule at $n = -1$: $\int \frac{dx}{x} = \ln|x|$, not $\frac{x^0}{0}$.
- Subtracting in the wrong order in a definite integral: the upper limit first, then the lower.