Calculus Examples

$\int \frac{3 x + 2}{\sqrt{1 - x^{2}}} d x$

Step 1

Let $x = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$\int \frac{3 \sin (t) + 2}{\sqrt{1 - \sin^{2} (t)}} \cos (t) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{1 - \sin^{2} (t)}$ .

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Step 2.1.1

Apply pythagorean identity.

$\int \frac{3 \sin (t) + 2}{\sqrt{\cos^{2} (t)}} \cos (t) d t$

Step 2.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\int \frac{3 \sin (t) + 2}{\cos (t)} \cos (t) d t$

Step 2.2

Cancel the common factor of $\cos (t)$ .

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Step 2.2.1

Cancel the common factor.

$\int \frac{3 \sin (t) + 2}{\cos (t)} \cos (t) d t$

Step 2.2.2

Rewrite the expression.

$\int 3 \sin (t) + 2 d t$

Step 3

Split the single integral into multiple integrals.

$\int 3 \sin (t) d t + \int 2 d t$

Step 4

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int \sin (t) d t + \int 2 d t$

Step 5

The integral of $\sin (t)$ with respect to $t$ is $- \cos (t)$ .

$3 (- \cos (t) + C) + \int 2 d t$

Step 6

Apply the constant rule.

$3 (- \cos (t) + C) + 2 t + C$

Step 7

Simplify.

$- 3 \cos (t) + 2 t + C$

Step 8

Replace all occurrences of $t$ with $\arcsin (x)$ .

$- 3 \cos (\arcsin (x)) + 2 \arcsin (x) + C$