Mathematics
Integrals
Integration is the reverse of differentiation — find the function whose derivative is the one you're given. Most integrals need a substitution or a technique like integration by parts.
How to Approach Integrals
Recognize the form
Is it a polynomial, a composition (try u-substitution), a product (try parts), or a trig expression (try a trig identity or trig sub)?
Apply the technique
u-sub: let u = inner function, du = its derivative. Parts: ∫u dv = uv - ∫v du. Trig sub: replace x with sin/tan/sec depending on form.
Back-substitute
After integrating in u, replace u with the original expression. For definite integrals, evaluate at the bounds and subtract.
Frequently Asked Questions
What's the difference between definite and indefinite?+
Indefinite integrals return a function plus a constant: ∫f(x)dx = F(x) + C. Definite integrals return a number: ∫ₐᵇf(x)dx = F(b) - F(a).
When do I use integration by parts?+
When the integrand is a product of two functions where one becomes simpler when differentiated. Classic example: ∫x·eˣ dx — let u = x, dv = eˣ dx.
Why does u-substitution work?+
It's the chain rule run backwards. By substituting u for the inner function, you convert a composition into a simpler integral you can solve directly.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
Stuck on a Integrals problem?
Snap a photo or type the question. ScanSolve walks you through every step — same as the worked examples above. 5 free solves per day, no card required.