Math / Calculus

Question4. Compute the indefinite integral $\int \sec ^{3} x d x$ .

Studdy Solution

STEP 1

What is this asking? We need to find the family of functions whose derivative is $\sec^3(x)$ . Watch out! This integral requires a clever trick using integration by parts, and it's easy to get lost in the algebra.

STEP 2

1. Set up Integration by Parts
2. Solve for the Integral
3. Simplify the Result

STEP 3

Alright, let's tackle this integral using integration by parts!
Remember the formula: $\int u dv = uv - \int v du$ .
Now, we need to choose our u and dv wisely.
Let's set $u = \sec(x)$ and $dv = \sec^2(x) dx$ .

STEP 4

Why this choice?
Well, if $u = \sec(x)$ , then $du = \sec(x) \cdot \tan(x) dx$ .
And if $dv = \sec^2(x) dx$ , then $v = \tan(x)$ .
See how nicely that works out?

STEP 5

Plugging these into our integration by parts formula, we get: $\int \sec^3(x) dx = \sec(x) \cdot \tan(x) - \int \tan(x) \cdot \sec(x) \cdot \tan(x) dx$ $\int \sec^3(x) dx = \sec(x) \cdot \tan(x) - \int \sec(x) \cdot \tan^2(x) dx$

STEP 6

Now, remember the trigonometric identity: $\tan^2(x) = \sec^2(x) - 1$ .
Let's substitute that in: $\int \sec^3(x) dx = \sec(x) \cdot \tan(x) - \int \sec(x) \cdot (\sec^2(x) - 1) dx$ $\int \sec^3(x) dx = \sec(x) \cdot \tan(x) - \int (\sec^3(x) - \sec(x)) dx$ $\int \sec^3(x) dx = \sec(x) \cdot \tan(x) - \int \sec^3(x) dx + \int \sec(x) dx$

STEP 7

Look! We have $\int \sec^3(x) dx$ on both sides of the equation.
Let's add $\int \sec^3(x) dx$ to both sides: $2 \int \sec^3(x) dx = \sec(x) \cdot \tan(x) + \int \sec(x) dx$

STEP 8

We know that $\int \sec(x) dx = \ln|\sec(x) + \tan(x)| + C$ .
So, we have: $2 \int \sec^3(x) dx = \sec(x) \cdot \tan(x) + \ln|\sec(x) + \tan(x)| + C$

STEP 9

Finally, we divide both sides by $2$ to isolate our integral: $\int \sec^3(x) dx = \frac{1}{2} (\sec(x) \cdot \tan(x) + \ln|\sec(x) + \tan(x)|) + C$ Remember, since C is an arbitrary constant, $\frac{C}{2}$ is also an arbitrary constant, so we can just keep calling it C.

STEP 10

The indefinite integral is: $\int \sec^3(x) dx = \frac{1}{2} (\sec(x) \cdot \tan(x) + \ln|\sec(x) + \tan(x)|) + C$

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