Answer to Question #336648 in Calculus for Kayle

Question

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

Answer 1

1)

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

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

2)

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

$$=4x^3+x^2+C$$

3)

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

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

4)

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

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

5)

$$\int \sqrt[3]{x}dx=\int x^{1/3}dx=\dfrac{x^{4/3}}{4/3}+C$$

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