Question

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C

Graph the following functions and find the area under the curve on the given interval.
f(x) = (9 - x²) ⁻², [0, 3/2]

Accepted Answer

Identify the integral you need to evaluate: the area under the curve of the function $$f(x) = (9 - x$$^{2}$$)^{-2} on $$the interval $$[0, \frac{3}{2}]. $$This means you want to compute the definite integral $$\int_{0}^{\frac{3}{2}} (9 - x$$^{2}$$)^{-2}$$ \, dx$$. Consider a substitution to simplify the integral. Since the integrand involves 9$$ - $$x^{2}$$, use the substitution $$x = 3 \sin u$$, which implies $$dx = 3 \cos u \, du. $$This substitution transforms the integral into terms of $$u. $$Rewrite the integral in terms of $$u$$: replace $$x$$ and $$dx$$ accordingly, and express the limits of integration in terms of $$u. $$When $$x=0$$, $$u=\arcsin(0)=0$$; when $$x=\frac{3}{2}$$, $$u=\arcsin(\frac{1}{2})=\frac{\pi}{6}. $$Simplify the integrand after substitution. Note that $$9 - x$$^{2}$$ = 9 - 9 \sin$$^{2}$$ u = 9 \cos$$^{2}$$ u$$, so $$(9 - x$$^{2}$$)^{-2} = (9$$ \$$cos^{2}$$ $$u)^{-2}$$ = \frac{1}{81 \$$cos^{4} u$$}$$. Incorporate dx = 3$$ \cos u \, du$$ to rewrite the integral as $$\int_{0}^{\frac{\pi}{6}} \frac{3 \cos u}{81 \$$cos^{4} u$$} \, du = \int_{0}^{\frac{\pi}{6}} \frac{3}{81 \$$cos^{3} u$$} \, du$$. Recognize that $$\frac{1}{\$$cos^{3} u$$} = \$$sec^{3} u$, so the integral becomes $$\frac{1}{27} \int_{0}^{\frac{\pi}{6}} \$$sec^{3} u$$ \, du$$. Use the given reduction formula for $$\int \$$sec^{3} u$$ \, du = \frac{1}{2} (\sec u 	an u + \ln |\sec u + 	an u|) + C$$ to evaluate this definite integral between 0$ and $$\frac{\pi}{6}$.

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C

Graph the following functions and find the area under the curve on the given interval.
f(x) = (9 - x²) ⁻², [0, 3/2]

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Integration of Trigonometric Functions Using Reduction Formulas

Definite Integrals and Area Under a Curve

Graphing Functions to Understand Behavior

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + CGraph the following functions and find the area under the curve on the given interval.f(x) = (9 - x²) ⁻², [0, 3/2]

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Integration of Trigonometric Functions Using Reduction Formulas

Definite Integrals and Area Under a Curve

Graphing Functions to Understand Behavior

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C

Graph the following functions and find the area under the curve on the given interval.
f(x) = (9 - x²) ⁻², [0, 3/2]