Understanding antiderivatives and fundamental integration techniques
Area above x-axis: positive contribution
Area below x-axis: negative contribution
\[\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx\]
Reversing limits negates the integral
\[\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx\]
Additivity property
\[\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\]
\[\int e^x\,dx = e^x + C\]
\[\int \frac{1}{x}\,dx = \ln|x| + C\]
\[\int \sin(x)\,dx = -\cos(x) + C\]
\[\int \cos(x)\,dx = \sin(x) + C\]
Using substitution to simplify complex integrals
Identify u = g(x) where g'(x) appears in the integrand
Express du = g'(x)dx
Substitute and integrate in terms of u
Substitute back to x
Evaluate \[\int x e^{x^2}\,dx\]
Let u = x²
Then du = 2x dx
\[\frac{1}{2}\int e^u\,du\]
\[\frac{1}{2}e^u + C\]
\[\frac{1}{2}e^{x^2} + C\]
A technique based on the product rule
\[\int u\,dv = uv - \int v\,du\]
Evaluate \[\int x\ln(x)\,dx\]
u = ln(x)
dv = x dx
du = \(\frac{1}{x}dx\)
v = \(\frac{x^2}{2}\)
\[\frac{x^2}{2}\ln(x) - \int \frac{x^2}{2} \cdot \frac{1}{x}dx\]
\[\frac{x^2}{2}\ln(x) - \frac{x^2}{4} + C\]
Test your understanding of integration techniques.