4. express the integral:\n\n\\( \\int _ { 0 } ^ { 1 } \\int _ { 2 } ^ { 3 } \\int _ { 4 } ^ { 5 } \\sin ( x…

4. express the integral:\n\n\\( \\int _ { 0 } ^ { 1 } \\int _ { 2 } ^ { 3 } \\int _ { 4 } ^ { 5 } \\sin ( x ^ { 2 } ) \\cos ( y ^ { 3 } ) \\ln ( 1 + z ^ { 4 } ) d z d y d x \\)\n\nas a product of integrals of one variable. do not attempt to evaluate those integrals!

4. express the integral:\n\n\\( \\int _ { 0 } ^ { 1 } \\int _ { 2 } ^ { 3 } \\int _ { 4 } ^ { 5 } \\sin ( x ^ { 2 } ) \\cos ( y ^ { 3 } ) \\ln ( 1 + z ^ { 4 } ) d z d y d x \\)\n\nas a product of integrals of one variable. do not attempt to evaluate those integrals!

Answer

Explanation:

Step1: Analyze the limits of integration

The given integral is a triple - integral (\int_{0}^{1}\int_{2}^{3}\int_{4}^{5}\sin(x^{2})\cos(y^{3})\ln(1 + z^{4})dzdydx).

Step2: Use the property of multiple - integrals

For a triple - integral (\int_{a}^{b}\int_{c}^{d}\int_{e}^{f}f(x)g(y)h(z)dzdydx), we can use the property (\int_{a}^{b}\int_{c}^{d}\int_{e}^{f}f(x)g(y)h(z)dzdydx=\left(\int_{a}^{b}f(x)dx\right)\left(\int_{c}^{d}g(y)dy\right)\left(\int_{e}^{f}h(z)dz\right))

Here, (f(x)=\sin(x^{2})), (g(y)=\cos(y^{3})), (h(z)=\ln(1 + z^{4}))

So, (\int_{0}^{1}\int_{2}^{3}\int_{4}^{5}\sin(x^{2})\cos(y^{3})\ln(1 + z^{4})dzdydx=\int_{0}^{1}\sin(x^{2})dx\int_{2}^{3}\cos(y^{3})dy\int_{4}^{5}\ln(1 + z^{4})dz)

Answer:

(\int_{0}^{1}\sin(x^{2})dx\int_{2}^{3}\cos(y^{3})dy\int_{4}^{5}\ln(1 + z^{4})dz)