int 2x cos^3(x^2) sin(x^2) , dx = | vedclass

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Question

(A) Let $t = \cos(x^2)$.Then,differentiating with respect to $x$,we get $dt = -\sin(x^2) \cdot 2x \, dx$,which implies $2x \sin(x^2) \, dx = -dt$.Subs

Vedclass · Accepted Answer

$-\frac{1}{4} \cos^4(x^2) + c$

Vedclass · Answer

(A) Let $t = \cos(x^2)$.Then,differentiating with respect to $x$,we get $dt = -\sin(x^2) \cdot 2x \, dx$,which implies $2x \sin(x^2) \, dx = -dt$.Substituting these into the integral:$\int 2x \cos^3(x^2) \sin(x^2) \, dx = \int t^3 (-dt)$$= -\int t^3 \, dt$$= -\frac{t^4}{4} + c$Substituting $t = \cos(x^2)$ back into the expression:$= -\frac{1}{4} \cos^4(x^2) + c$.

$\int 2x \cos^3(x^2) \sin(x^2) \, dx = $

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