← Integral Calculus
Mathematics Β· Topic Cheatsheet

Integral Calculus

16 key results accumulated across 2 chapters.

Power rule (reverse)
Ch 1
∫xn dx=xn+1n+1+Cβ€…β€Š(nβ‰ βˆ’1)\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\;(n\neq-1)
Exponential / reciprocal
Ch 1
∫exdx=ex+C,∫1x dx=ln⁑∣x∣+C\int e^x dx = e^x + C,\quad \int \tfrac{1}{x}\,dx = \ln|x| + C
Definite integral (FTC)
Ch 1
∫abf(x) dx=F(b)βˆ’F(a)\int_a^b f(x)\,dx = F(b) - F(a)
+C reminder
Ch 1
Indefinite integrals always carry the constant of integration +C+C.
Power rule (integral)
Ch 1
∫xn dx=xn+1n+1+Cβ€…β€Š(nβ‰ βˆ’1)\int x^n\,dx = \tfrac{x^{n+1}}{n+1}+C\;(n\neq-1)
+C and definite
Ch 1
∫abf=F(b)βˆ’F(a)\int_a^b f = F(b)-F(a)
Indefinite needs +C; definite gives a number (no C).
Standard integrals
Ch 1
∫ex=ex,β€…β€Šβˆ«1x=ln⁑∣x∣,β€…β€Šβˆ«cos⁑x=sin⁑x\int e^x = e^x,\;\int\tfrac1x = \ln|x|,\;\int\cos x = \sin x
Common trap
Ch 1
Don’t forget +C; ∫1/x = ln|x| (absolute value).
Substitution (reverse chain rule)
Ch 2
∫f(g(x))gβ€²(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx = \int f(u)\,du
Substitution steps
Ch 2
Let uu = inner function; replace gβ€²(x)dxg'(x)dx with dudu; integrate in uu; substitute back.
By parts (reverse product rule)
Ch 2
∫u dv=uvβˆ’βˆ«v du\int u\,dv = uv - \int v\,du
Choosing u (LIATE)
Ch 2
Log, Inverse-trig, Algebraic, Trig, Exponential β€” pick uu earliest in this list.
Common results
Ch 2
∫xexdx=(xβˆ’1)ex+C,∫xcos⁑x dx=xsin⁑x+cos⁑x+C\int xe^x dx = (x-1)e^x + C,\quad \int x\cos x\,dx = x\sin x + \cos x + C
Substitution
Ch 2
∫f(g(x))gβ€²(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx = \int f(u)\,du
Change limits too for definite integrals.
By parts
Ch 2
∫u dv=uvβˆ’βˆ«v du\int u\,dv = uv - \int v\,du
Pick u by LIATE (log, inverse-trig, algebraic, trig, exp).
Common trap
Ch 2
After substitution, convert limits to u (or back-substitute before applying x-limits).