Question #336648

Topic: Integration



Find the antiderivative of the following. Show your comple solution.



1.) Integral (6x⁵-4x²+2) dx


2.) Integral (12x²+2x) dx


3.) Integral 8^3/2 dx


4.) Integral 1/x¹² dx


5.) Integral ³√x dx

1
Expert's answer
2022-05-03T16:41:34-0400

1)

(6x54x3+2)dx=6(x66)4(x44)+2x+C\int(6x^5-4x^3+2) dx=6(\dfrac{x^6}{6})-4(\dfrac{x^4}{4})+2x+C

=x6x4+2x+C=x^6-x^4+2x+C

2)

(12x2+2x)dx=12(x33)+2(x22)+C\int(12x^2+2x) dx=12(\dfrac{x^3}{3})+2(\dfrac{x^2}{2})+C

=4x3+x2+C=4x^3+x^2+C

3)

(83/2)dx=83/2dx\int(8^{3/2}) dx=8^{3/2}\int dx

=83/2x+C=8^{3/2}x+C

4)

(1x12)dx=x12dx=x1111+C\int(\dfrac{1}{x^{12}}) dx=\int x^{-12}dx=\dfrac{x^{-11}}{-11}+C

=111x11+C=-\dfrac{1}{11x^{11}}+C

5)

x3dx=x1/3dx=x4/34/3+C\int \sqrt[3]{x}dx=\int x^{1/3}dx=\dfrac{x^{4/3}}{4/3}+C

=3xx34+C=\dfrac{3x\sqrt[3]{x}}{4}+C

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