Integrating powers of (ax + b) questions with answers

Questions

1. Question 1(ax + b) to a whole power

   Find $\int 2(4x + 3)^{5} \, dx$.

   Show the worked solution

   1. Step 1: Add 1 to the power, then divide by the new power AND by the coefficient of x$= \dfrac{2(4x + 3)^{6}}{6 \times 4} + c$
   2. Step 2: Simplify the number in front$= \dfrac{1}{12}(4x + 3)^{6} + c$
2. Question 2(ax + b) to a whole power

   Find $\int 6(3x - 1)^{4} \, dx$.

   Show the worked solution

   1. Step 1: Add 1 to the power, then divide by the new power AND by the coefficient of x$= \dfrac{6(3x - 1)^{5}}{5 \times 3} + c$
   2. Step 2: Simplify the number in front$= \dfrac{2}{5}(3x - 1)^{5} + c$
3. Question 3Negative powers and roots

   Find $\int 3\sqrt{3x - 1} \, dx$.

   Show the worked solution

   1. Step 1: Write it as a power of the bracket$\int 3(3x - 1)^{\frac{1}{2}} \, dx$
   2. Step 2: Add 1 to the power, then divide by the new power AND by the coefficient of x$= \dfrac{3(3x - 1)^{\frac{3}{2}}}{\left(\frac{3}{2}\right) \times 3} + c$
   3. Step 3: Simplify the number in front$= \dfrac{2}{3}(3x - 1)^{\frac{3}{2}} + c$
4. Question 4Negative powers and roots

   Find $\int \dfrac{4}{\sqrt{x + 7}} \, dx$.

   Show the worked solution

   1. Step 1: Write it as a power of the bracket$\int 4(x + 7)^{-\frac{1}{2}} \, dx$
   2. Step 2: Add 1 to the power, then divide by the new power AND by the coefficient of x$= \dfrac{4(x + 7)^{\frac{1}{2}}}{\left(\frac{1}{2}\right) \times 1} + c$
   3. Step 3: Simplify the number in front$= 8(x + 7)^{\frac{1}{2}} + c$