Question

1. (5 points) which of the following is equal to the integral (int\frac{dx}{(81 - x^{2})^{\frac{3}{2}}}) after making the substitution (x = 9sin(\theta))? a. (int\frac{d\theta}{729cos^{3}(\theta)}) b. (int\frac{cos(\theta)d\theta}{81 - 81sin^{3}(\theta)}) c. (int\frac{1}{81}sec^{2}(\theta)d\theta) d. (int\frac{1}{81}\tan^{2}(\theta)d\theta) e. (int\frac{d\theta}{729 - 729sin^{3}(\theta)})

Explanation:

Step1: Find $dx$

If $x = 9\sin(\theta)$, then $dx=9\cos(\theta)d\theta$.

Step2: Substitute $x$ in the denominator

Substitute $x = 9\sin(\theta)$ into $81 - x^{2}$. We get $81-81\sin^{2}(\theta)=81(1 - \sin^{2}(\theta))$. Since $1-\sin^{2}(\theta)=\cos^{2}(\theta)$, then $(81 - x^{2})^{\frac{3}{2}}=(81\cos^{2}(\theta))^{\frac{3}{2}}=729\cos^{3}(\theta)$.

Step3: Rewrite the integral

The original integral $\int\frac{dx}{(81 - x^{2})^{\frac{3}{2}}}$ becomes $\int\frac{9\cos(\theta)d\theta}{729\cos^{3}(\theta)}=\int\frac{d\theta}{81\cos^{2}(\theta)}=\int\frac{1}{81}\sec^{2}(\theta)d\theta$.

Answer:

C. $\int\frac{1}{81}\sec^{2}(\theta)d\theta$