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int tsqrt(t^(2)+2)dt

$\int t \sqrt{t^{2}+2} d t$

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Find indefinite integrals using the substitution

Full solution

Q.

\int t \sqrt{t^{2}+2} d t

Make Substitution: Let's start by making a substitution to simplify the integral. Let $u = t^2 + 2$ . Then, $du = 2t dt$ , or $dt = \frac{du}{2t}$ .
Substitute and Simplify: Substitute into the integral: $\int t\sqrt{t^2 + 2} \, dt = \int t\sqrt{u} \cdot \left(\frac{du}{2t}\right)$ . The $t$ 's cancel out, so we get: $\frac{1}{2} \int \sqrt{u} \, du$ .
Integrate $u^{1/2}$ : Now, integrate $u^{1/2}$ . The integral of $u^{1/2}$ is $(2/3)u^{3/2} + C$ .
Substitute Back: Substitute back for $u$ : $\left(\frac{2}{3}\right)\left(t^2 + 2\right)^{\frac{3}{2}} + C$ .

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