Test: Integration Basics - JEE MCQ

Step 1: Understand the Problem 

• The problem is asking for the integral of the function 7cos(mx) with respect to x.

Step 2: Apply the Integral Rule

• The integral of cos(mx) is (1/m)sin(mx).
• Since there is a constant multiplier, 7, the constant can be factored out of the integral.

Step 3: Solve the Integral

• Apply the integral rule to cos(mx) and multiply the result by the constant 7.

Step 4: Write the Final Answer

• The integral of 7cos(mx) dx is 7*(1/m)sin(mx) + C, where C is the constant of integration.

Final Answer:

• The integral ∫7cos(mx) dx = (7/m)sin(mx) + C

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.