Question

Find \(\int_0^π(1-sin⁡3x)dx\).

(a) \(\frac{3π-2}{4}\)

(b) 3π-1

(c) \(\frac{3π-2}{3}\)

(d) π-\(\frac{1}{3}\)

This question was addressed to me in quiz.

My query is from Fundamental Theorem of Calculus-2 in division Integrals of Mathematics – Class 12

Answer 1

Right option is (c) \(\frac{3π-2}{3}\)

The explanation is: Let \(I=\int_0^π(1-sin⁡3x)dx\)

F(x)=∫ 1-sin⁡3x dx

=x+\(\frac{cos⁡3x}{3}\)

Applying the limits by using the fundamental theorem of calculus, we get

I=F(π)-F(0)

=\(π+\frac{cos⁡3π}{3}-0-\frac{cos⁡0}{3}\)

=\(π-\frac{1}{3}-\frac{1}{3}=π-\frac{2}{3}=\frac{3π-2}{3}\).