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Integration by Parts Definite Integral

Let u = f(z) and g(z) be functions with continuous derivatives. Then, the integration-by-parts formula for the integral involving these two functions is u*dv = f(z)*g(z) – âˆ«g(z)*du.

Evaluate the indefinite integral âˆ«âˆš(x)dx. Note: type an exact answer without using decimals.

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04:16

Integration by Parts

03:17

Express each integral as an equivalent integral in which the $z$ -integration is performed first, the $y$ -integration second, and the $x$ -integration last.$$\begin{array}{l}{\text { (a) } \int_{0}^{3} \int_{0}^{\sqrt{9-z^{2}}} \int_{0}^{\sqrt{9-y^{2}-z^{2}}} f(x, y, z) d x d y d z} \\ {\text { (b) } \int_{0}^{4} \int_{0}^{2} \int_{0}^{x / 2} f(x, y, z) d y d z d x} \\ {\text { (c) } \int_{0}^{4} \int_{0}^{4-y} \int_{0}^{\sqrt{z}} f(x, y, z) d x d z d y}\end{array}$$

02:34

Express each integral as an equivalent integral in which the $z$ -integration is performed first, the $y$ -integration second, and the $x$ -integration last.$$\begin{array}{l}{\text { (a) } \int_{0}^{5} \int_{0}^{2} \int_{0}^{\sqrt{4-y^{2}}} f(x, y, z) d x d y d z} \\ {\text { (b) } \int_{0}^{9} \int_{0}^{3-\sqrt{x}} \int_{0}^{z} f(x, y, z) d y d z d x} \\ {\text { (c) } \int_{0}^{4} \int_{y}^{8-y} \int_{0}^{\sqrt{4-y}} f(x, y, z) d x d z d y}\end{array}$$

03:42

Evaluate the integrals by using a substitution prior to integration by parts.$$\int z(\ln z)^{2} d z$$

07:58

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