Evaluate the Integral

Problem

(∫_0^π)((6+sin(x))*d(x))

Solution

1. Split the integral into two separate terms using the linearity property of integration.

(∫_0^π)(6*d(x))+(∫_0^π)(sin(x)*d(x))

2. Find the antiderivative of each term. The antiderivative of 6 is 6*x and the antiderivative of sin(x) is −cos(x)

[6*x]π0+[−cos(x)]π0

3. Apply the Fundamental Theorem of Calculus to the first term by evaluating the antiderivative at the upper and lower limits.

6*(π)−6*(0)=6*π

4. Apply the Fundamental Theorem of Calculus to the second term.

(−cos(π))−(−cos(0))

5. Evaluate the trigonometric values where cos(π)=−1 and cos(0)=1

−(−1)−(−1)=1+1=2

6. Combine the results of both evaluated parts to find the final value.

6*π+2

Final Answer

(∫_0^π)((6+sin(x))*d(x))=6*π+2

Want more problems? Check here!