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10 cards from this deck
To integrate composite functions effectively.
Identify the inner function in the integrand.
Set u=g(x)u = g(x)u=g(x) where g(x)g(x)g(x) is the inner function.
Differentiate uuu with respect to xxx to find du/dxdu/dxdu/dx.
Replace g(x)g(x)g(x) with uuu and dxdxdx with du/(du/dx)du/(du/dx)du/(du/dx).
The integral should now be in terms of uuu and dududu.
Substitute back u=g(x)u = g(x)u=g(x) into the integral's result.
Inner function is x2x^2x2, with g′(x)=2xg'(x) = 2xg′(x)=2x.
Integral of cos(u)\cos(u)cos(u) is sin(u)+C\sin(u) + Csin(u)+C.
The integral is ex2+Ce^{x^2} + Cex2+C by u-substitution.
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