alleryFind $\displaystyle\int 2x \cos(x^2)\,dx$.
A$2 \cos(x^2) + C$
B$\sin(x^2) + C$
C$\dfrac{\sin(x^2)}{2x} + C$
D$x^2 \sin(x^2) + C$
Answer & Solution
Correct answer: B. $\sin(x^2) + C$
Use substitution $u = x^2 \Rightarrow du = 2x\,dx$. The $2x\,dx$ in the integrand is exactly $du$, so the integral becomes:
$\displaystyle\int \cos u\,du = \sin u + C = \sin(x^2) + C$.
This pattern (the derivative of an inner function appearing as a factor) is the textbook cue for substitution.
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