Calculus

Integration Techniques

Integration U-substitution - Problem Solving

         

When a=2a=2 and b=4,b=4, ab(xa)(bx)dx \int_{a}^{b}{\sqrt{(x-a)(b-x)}}dx is equal to .\underline{\qquad}.

For nn a positive integer, what is the indefinite integral x(1x)n+6dx?\int{ x{(1-x)}^{n+6} }dx?

Details and assumptions

Use CC as the constant of integration.

Evaluate 03(x+4)2ex2dx+30(x4)2ex2dx.\int_{0}^{\sqrt{3}} (x+4)^2 e^{x^2} dx+\int_{\sqrt{3}}^{0} (x-4)^2 e^{x^2} dx.

Given that 04x39+x2dx=a\displaystyle \int_0^4 x^3\sqrt{9+x^2} dx = a, what is the value of a\lfloor a \rfloor?

Details and assumptions

Greatest Integer Function: x:RZ\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z} refers to the greatest integer less than or equal to xx. For example 2.3=2\lfloor 2.3 \rfloor = 2 and 3.4=4\lfloor -3.4 \rfloor = -4.

Determine the value of 1000π1+sinxdx 100 \int_0^{\pi} \sqrt{ 1 + \sin x} \, dx .

This problem is proposed by Cody.

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