Test: Integration Basics - JEE MCQ

The problem is asking to find the integral of 6x(x²+6)dx. This is a straightforward application of the power rule for integration.

Step 1:
First, distribute the 6x into the bracket.

• 6x * x² + 6x * 6 = 6x³ + 36x

Step 2:
Now, we can find the integral of each term separately.

• ∫6x³ dx + ∫36x dx

Step 3:
Apply the power rule for integration, which states that the integral of x^n dx is (1/n+1)x^(n+1).

• (6/4)x⁴ + (36/2)x² = 1.5x⁴ + 18x²

Step 4:
Finally, don't forget to add the constant of integration, denoted as 'c'.

Therefore, the integral of 6x(x²+6)dx is 1.5x⁴ + 18x² + c.

So, according to the options provided, the correct answer is Option C: f(x)+c.

Method1: by cross checking the options

Hence option 3 is the correct answer and we can check the other choices by the similar argument.