For which of the following functions is the s | vedclass

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(C) The substitution ${x^2} = t$ implies $2x \,dx = dt$,or $x \,dx = \frac{1}{2} \,dt$.For option $(c)$,we have $\int {x^3 \cos(x^2)} \,dx = \int {x^2

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$\int {x^3 \cos(x^2)} \,dx$

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(C) The substitution ${x^2} = t$ implies $2x \,dx = dt$,or $x \,dx = \frac{1}{2} \,dt$.For option $(c)$,we have $\int {x^3 \cos(x^2)} \,dx = \int {x^2 \cdot \cos(x^2) \cdot x \,dx}$.Substituting ${x^2} = t$ and $x \,dx = \frac{1}{2} \,dt$,we get $\frac{1}{2} \int {t \cos t \,dt}$.Using integration by parts: $\frac{1}{2} [t \sin t - \int \sin t \,dt] = \frac{1}{2} (t \sin t + \cos t) + C$.Substituting back $t = x^2$,we get $\frac{1}{2} (x^2 \sin(x^2) + \cos(x^2)) + C$.Thus,option $(c)$ is correct.

For which of the following functions is the substitution ${x^2} = t$ applicable?

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