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intx^((1)/(3))cos(x^(2//3),8)dx

$\int x^{\frac{1}{3}} \cos \left(x^{2 / 3}, 8\right) d x$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\int x^{\frac{1}{3}} \cos \left(x^{2 / 3}, 8\right) d x

Substitution step: Let's do a substitution: let $u = x^{\frac{2}{3}}$ . Then, $du = \left(\frac{2}{3}\right)x^{-\frac{1}{3}}dx$ .
Finding dx: To find dx, we rearrange the equation: $dx = \left(\frac{3}{2}\right)x^{\frac{1}{3}}du$ .
Changing limits of integration: Now we need to change the limits of integration. When $x = 0$ , $u = 0^{(2/3)} = 0$ . When $x = 8$ , $u = 8^{(2/3)} = 4$ .
Final substitution into integral: Substitute everything into the integral: $\int x^{\frac{1}{3}}\cos(x^{\frac{2}{3}})dx = \int \cos(u) \cdot \left(\frac{3}{2}\right)u^{\frac{3}{2}}du$ from $0$ to $4$ .

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