MT2-2 a) $$ \int \frac{1}{\sqrt{9-25 x^{2}}} d x $$ AmDCrath Answer any 1 questions from 5 to 6 . Each carries 4 Marks. 5 a) Find the area enclosed by the circle $$ x^{2}+y^{2}=25 $$ using integration. b) If $$ \int \frac{f(x)}{ an x} d x=\log | an x|+C $$, then $$ f(x)=\ldots $$. 6. a) Find the area of the region bounded by $$ y^{2}=9 x, x=2, x=4 $$ and the $$ x $$-axis in the first quadrant. b) Find $$ \int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x $$ a) Find $$ \mathrm{y}^{2}=9 \mathrm{x} n $$ - (3) (1) b) $$ \int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x $$ किणनाइक Answer any 2 questions from 7 to 9 . Each carries 6 Marks . 7. a) The length $$ x $$ of a rectangle is increasing at the rate of $$ 4 \mathrm{~cm} / \mathrm{s} $$ and the width $$ y $$ is decreasing at the rate of $$ 5 \mathrm{~cm} / \mathrm{s} $$. When $$ \mathrm{x}=10 \mathrm{~cm} $$ and $$ \mathrm{y}=5 \mathrm{~cm} $$. i) Find the rates of change of its area. ii) Find the rates of change of the perimeter of the rectangle. $$ (2 $$ $$ (2 $$

Question

MT2-2 a) $$ \int \frac{1}{\sqrt{9-25 x^{2}}} d x $$ AmDCrath Answer any 1 questions from 5 to 6 . Each carries 4 Marks. 5 a) Find the area enclosed by the circle $$ x^{2}+y^{2}=25 $$ using integration. b) If $$ \int \frac{f(x)}{	an x} d x=\log |	an x|+C $$, then $$ f(x)=\ldots $$. 6. a) Find the area of the region bounded by $$ y^{2}=9 x, x=2, x=4 $$ and the $$ x $$-axis in the first quadrant. b) Find $$ \int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x $$ a) Find $$ \mathrm{y}^{2}=9 \mathrm{x} n $$ - (3) (1) b) $$ \int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x $$ किणनाइक Answer any 2 questions from 7 to 9 . Each carries 6 Marks . 7. a) The length $$ x $$ of a rectangle is increasing at the rate of $$ 4 \mathrm{~cm} / \mathrm{s} $$ and the width $$ y $$ is decreasing at the rate of $$ 5 \mathrm{~cm} / \mathrm{s} $$. When $$ \mathrm{x}=10 \mathrm{~cm} $$ and $$ \mathrm{y}=5 \mathrm{~cm} $$. i) Find the rates of change of its area. ii) Find the rates of change of the perimeter of the rectangle. $$ (2 $$ $$ (2 $$

UpStudy AI · Accepted Answer

To evaluate the integral:

$$
\int \frac{1}{\sqrt{9 - 25x^{2}}} \, dx
$$

we can use a substitution method involving inverse trigonometric functions.

### Step 1: Recognize the Integral Form

The integral resembles the standard form:

$$
\int \frac{1}{\sqrt{a^{2} - b^{2}x^{2}}} \, dx = \frac{1}{b} \sin^{-1}\left(\frac{b x}{a}\right) + C
$$

where $$ a = 3 $$ and $$ b = 5 $$.

### Step 2: Apply Substitution

Let’s set:

$$
u = \frac{5x}{3} \quad \Rightarrow \quad x = \frac{3u}{5} \quad \Rightarrow \quad dx = \frac{3}{5} du
$$

Substituting these into the integral:

$$
\int \frac{1}{\sqrt{9 - 25x^{2}}} \, dx = \int \frac{1}{\sqrt{9 - 25\left(\frac{3u}{5}\right)^{2}}} \cdot \frac{3}{5} \, du
$$

Simplify inside the square root:

$$
\sqrt{9 - 25 \cdot \frac{9u^{2}}{25}} = \sqrt{9 - 9u^{2}} = 3\sqrt{1 - u^{2}}
$$

So the integral becomes:

$$
\int \frac{1}{3\sqrt{1 - u^{2}}} \cdot \frac{3}{5} \, du = \frac{1}{5} \int \frac{1}{\sqrt{1 - u^{2}}} \, du
$$

### Step 3: Integrate

The integral of $$ \frac{1}{\sqrt{1 - u^{2}}} $$ is $$ \sin^{-1}(u) $$:

$$
\frac{1}{5} \sin^{-1}(u) + C
$$

### Step 4: Substitute Back

Recall that $$ u = \frac{5x}{3} $$:

$$
\frac{1}{5} \sin^{-1}\left(\frac{5x}{3}\right) + C
$$

### Final Answer

$$
\int \frac{1}{\sqrt{9 - 25x^{2}}} \, dx = \frac{1}{5} \sin^{-1}\left(\frac{5x}{3}\right) + C
$$

Where $$ C $$ is the constant of integration.
$$ \int \frac{1}{\sqrt{9 - 25x^{2}}} \, dx = \frac{1}{5} \sin^{-1}\left(\frac{5x}{3}\right) + C $$

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