Integral Calculus2D
Antiderivatives and Indefinite Integrals - Theory & Concepts - U Substitution
Understanding antiderivatives, indefinite integral notation, basic integration formulas, and properties of linearity.
Open the complete lessonU-Substitution: Coordinate Mapping & Area Equivalence
Visualizing the reverse Chain Rule. Watch how the coordinate stretching factor compresses/stretches the area elements between the original space and the substituted space.
x = 0x = 2
Coordinate Mapping: $u = x^3$
Mapped point $u$:1.7280
Stretch factor $du/dx = 3x^2$:4.320
∫ 3x² cos(x³) dx Area:0.98767
∫ cos(u) du Area:0.98767
The areas under both curves are exactly equal at every point! The $x$-space integrand is compressed on the left but stretched vertically by $3x^2$, matching the right $u$-space integral perfectly.
Original X-Space: $\int 3x^2\cos(x^3)dx$
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x-axis range [0, 2]
Transformed U-Space: $\int \cos(u)du$
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u-axis range [0, 8] ($u = x^3$)