$2$ . $\int_{0}^{\frac{1}{2} \pi^{2}} \cos \sqrt{2 t} d t$

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Evaluate definite integrals using the chain rule

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2

\int_{0}^{\frac{1}{2} \pi^{2}} \cos \sqrt{2 t} d t

Substitution: Let's do a substitution: $u = \sqrt{2t}$ , which means $t = \frac{u^2}{2}$ . Now we need to find $dt$ in terms of $du$ .
Differentiate to find $dt$ : Differentiate both sides with respect to $u$ to find $dt$ : $dt = \frac{u \, du}{\sqrt{2}}$ .
Change limits of integration: Now we change the limits of integration. When $t = 0$ , $u = \sqrt{2\cdot 0} = 0$ . When $t = \left(\frac{1}{2}\right)\pi^2$ , $u = \sqrt{2\cdot \left(\frac{1}{2}\right)\pi^2} = \pi$ .
Substitute into integral: Substitute everything into the integral: $\int_{0}^{\pi} \cos(u) \cdot \left(\frac{u}{\sqrt{2}}\right) \, du$ .
Simplify the integral: Simplify the integral: $\frac{1}{\sqrt{2}}$ $\int_{0}^{\pi} u \cos(u) \, du$ .
Integration by parts: This is an integral that requires integration by parts. Let's choose $v = u$ and $dw = \cos(u) du$ .
Find $dv$ and $w$ : Now we find $dv$ and $w$ . $dv = du$ , so $v = u$ . To find $w$ , we integrate $dw$ : $w = \int \cos(u) \, du = \sin(u)$ .
Apply integration by parts: Apply integration by parts: $(\frac{1}{\sqrt{2}}) \cdot [uv - \int v \, du]$ from $0$ to $\pi$ .
Evaluate definite integral: Plug in the values for $u$ and $w$ : $\frac{1}{\sqrt{2}}$ * $[u \sin(u) - \int u \, du]$ from $0$ to $\pi$ .
Evaluate at upper limit: Now we need to integrate $u \, \mathrm{d}u$ : $\int u \, \mathrm{d}u = \frac{u^2}{2}$ .
Evaluate at lower limit: Plug everything in and evaluate the definite integral: $(\frac{1}{\sqrt{2}}) \cdot [u \sin(u) - \frac{u^2}{2}]$ from $0$ to $\pi$ .
Subtract lower from upper: Evaluate at the upper limit: $(\frac{1}{\sqrt{2}}) * [\pi \sin(\pi) - \frac{\pi^2}{2}] = (\frac{1}{\sqrt{2}}) * [0 - \frac{\pi^2}{2}]$ .
Subtract lower from upper: Evaluate at the upper limit: $(1/\sqrt{2}) * [\pi \sin(\pi) - \pi^2/2] = (1/\sqrt{2}) * [0 - \pi^2/2]$ .Evaluate at the lower limit: $(1/\sqrt{2}) * [0 \sin(0) - 0^2/2] = 0$ .
Subtract lower from upper: Evaluate at the upper limit: $(1/\sqrt{2}) * [\pi \sin(\pi) - \pi^2/2] = (1/\sqrt{2}) * [0 - \pi^2/2]$ .Evaluate at the lower limit: $(1/\sqrt{2}) * [0 \sin(0) - 0^2/2] = 0$ .Subtract the lower limit from the upper limit: $(1/\sqrt{2}) * [-\pi^2/2] - 0 = -(\pi^2)/(2\sqrt{2})$ .