Question 22 of 25
What are the new limits of integration if apply the substitution $u=6 x+\pi$ to the integral $\int_{0}^{\pi} \sin (6 x+\pi) d x$ ?
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
lower limit:
upper limit:

Answer

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Answer

The new limits of integration are $\boxed{\pi}$ and $\boxed{7\pi}$.

Steps

Step 1 ：Apply the substitution $u=6 x+\pi$ to the integral $\int_{0}^{\pi} \sin (6 x+\pi) d x$.

Step 2 ：Substitute the lower limit of integration $x=0$ into $u=6 x+\pi$ to get the new lower limit. This gives $u=\pi$.

Step 3 ：Substitute the upper limit of integration $x=\pi$ into $u=6 x+\pi$ to get the new upper limit. This gives $u=7\pi$.

Step 4 ：The new limits of integration are $\boxed{\pi}$ and $\boxed{7\pi}$.