Integrating powers of x questions with answers

Questions

1. Question 1Polynomials

   Find $\int \left(3x^{2} - 4x - 1\right) \, dx$.

   Show the worked solution

   1. Step 1: Integrate each term: add 1 to the power, then divide by the new power (a number k integrates to kx). Add the constant c$\int \left(3x^{2} - 4x - 1\right) \, dx = \dfrac{3x^{3}}{3} - \dfrac{4x^{2}}{2} - x + c$$= x^{3} - 2x^{2} - x + c$
2. Question 2Polynomials

   Find $\int \left(3x^{2} + 2x + 7\right) \, dx$.

   Show the worked solution

   1. Step 1: Integrate each term: add 1 to the power, then divide by the new power (a number k integrates to kx). Add the constant c$\int \left(3x^{2} + 2x + 7\right) \, dx = \dfrac{3x^{3}}{3} + \dfrac{2x^{2}}{2} + 7x + c$
3. Question 3Negative and fractional powers

   Find $\int \left(\dfrac{6}{x^{4}} + 2x\right) \, dx$.

   Show the worked solution

   1. Step 1: Write every term as a power of x$\int \left(\dfrac{6}{x^{4}} + 2x\right) \, dx = \int \left(6x^{-4} + 2x\right) \, dx$
   2. Step 2: Integrate each term: add 1 to the power, then divide by the new power (a number k integrates to kx). Add the constant c$\int \left(6x^{-4} + 2x\right) \, dx = \dfrac{6x^{-3}}{-3} + \dfrac{2x^{2}}{2} + c$$= -2x^{-3} + x^{2} + c$
   3. Step 3: Write the answer with positive indices$= -\dfrac{2}{x^{3}} + x^{2} + c$
4. Question 4Negative and fractional powers

   Find $\int \left(x^{5} + \dfrac{6}{\sqrt{x}}\right) \, dx$.

   Show the worked solution

   1. Step 1: Write every term as a power of x$\int \left(x^{5} + \dfrac{6}{\sqrt{x}}\right) \, dx = \int \left(x^{5} + 6x^{-\frac{1}{2}}\right) \, dx$
   2. Step 2: Integrate each term: add 1 to the power, then divide by the new power (a number k integrates to kx). Add the constant c$\int \left(x^{5} + 6x^{-\frac{1}{2}}\right) \, dx = \dfrac{x^{6}}{6} + \dfrac{6x^{\frac{1}{2}}}{\dfrac{1}{2}} + c$$= \dfrac{1}{6}x^{6} + 12x^{\frac{1}{2}} + c$
   3. Step 3: Write the answer with positive indices$= \dfrac{1}{6}x^{6} + 12\sqrt{x} + c$
5. Question 5Expand or split first

   Find $\int \dfrac{x - 8}{\sqrt{x}} \, dx$.

   Show the worked solution

   1. Step 1: Split the fraction: divide each term on top by the bottom$\int \dfrac{x - 8}{\sqrt{x}} \, dx = \int \left(x^{\frac{1}{2}} - 8x^{-\frac{1}{2}}\right) \, dx$
   2. Step 2: Integrate each term: add 1 to the power, then divide by the new power (a number k integrates to kx). Add the constant c$\int \left(x^{\frac{1}{2}} - 8x^{-\frac{1}{2}}\right) \, dx = \dfrac{x^{\frac{3}{2}}}{\dfrac{3}{2}} - \dfrac{8x^{\frac{1}{2}}}{\dfrac{1}{2}} + c$$= \dfrac{2}{3}x^{\frac{3}{2}} - 16x^{\frac{1}{2}} + c$
   3. Step 3: Write the answer with positive indices$= \dfrac{2}{3}x\sqrt{x} - 16\sqrt{x} + c$
6. Question 6Expand or split first

   Find $\int \dfrac{2x - 6}{\sqrt{x}} \, dx$.

   Show the worked solution

   1. Step 1: Split the fraction: divide each term on top by the bottom$\int \dfrac{2x - 6}{\sqrt{x}} \, dx = \int \left(2x^{\frac{1}{2}} - 6x^{-\frac{1}{2}}\right) \, dx$
   2. Step 2: Integrate each term: add 1 to the power, then divide by the new power (a number k integrates to kx). Add the constant c$\int \left(2x^{\frac{1}{2}} - 6x^{-\frac{1}{2}}\right) \, dx = \dfrac{2x^{\frac{3}{2}}}{\dfrac{3}{2}} - \dfrac{6x^{\frac{1}{2}}}{\dfrac{1}{2}} + c$$= \dfrac{4}{3}x^{\frac{3}{2}} - 12x^{\frac{1}{2}} + c$
   3. Step 3: Write the answer with positive indices$= \dfrac{4}{3}x\sqrt{x} - 12\sqrt{x} + c$