Question

Integrate the following function:

Answer 1

Answer: a

Explanation:

First term

`\int √(10x-15) \mathrm{dx}`

We can use the substitution u=10x-15, meaning dx = 1/10 du.

`=(1)/(10) \int √(u) \mathrm{du}\\\\=(1)/(15)u^(3/2)+C\\\\=(1)/(15)(10x-15)^(3/2)+C`

Second term

`\int (10x-15)^(10) \mathrm{dx}`

We can use the substitution u=10x-15, meaning dx = 1/10 du.

`=(1)/(10) \int u^(10) \mathrm{du}\\\\=(1)/(110) u^(11)+C\\ \\ =(1)/(110)(10x-15)^(11)+C`

Third term

`\int (1)/(10x-15) \mathrm{dx}`

We can use the substitution u=10x-15, meaning dx = 1/10 du.

`=(1)/(10) \int (1)/(u) \mathrm{du}\\\\=(1)/(10) \ln u+C\\\\=(1)/(10) \ln(10x-15)+C`

Adding these results, we get the answer to be a.