sec cube dx
sec3 dx
You left out an x after sec^3
The integral ∫sec^3(x) dx represents the antiderivative of the function sec^3(x) with respect to x. To find the solution, we can use the method of integration known as trigonometric substitution.
Here's a step-by-step process to compute the integral of sec^3(x) dx:
Step 1: Express sec^3(x) in terms of sine and cosine.
We know that sec(x) = 1/cos(x). Rearranging this equation, we have cos(x) = 1/sec(x). Substituting this expression into the integral, we get:
∫(1/sec^3(x)) dx = ∫(cos^3(x)) dx.
Step 2: Use trigonometric substitution.
To proceed with trigonometric substitution, let's introduce a new variable:
Let's substitute cos(x) = u.
Then, we can differentiate both sides of the equation with respect to x, yielding:
-sin(x) dx = du.
Step 3: Substitute in the differential and solve for dx.
Rearranging the equation, we get:
dx = -du / sin(x).
Step 4: Substitute the trigonometric substitution expressions into the integral.
Using the substitutions from Steps 2 and 3, we can rewrite the integral as:
∫(u^3) * (-du / sin(x)).
Step 5: Simplify the integrand.
Considering that sin(x) = √(1 - cos^2(x)), we have:
∫(u^3) * (-du / sin(x)) = ∫(-u^3 / √(1 - u^2)) du.
Step 6: Integrate the simplified expression.
Now, we can integrate the expression ∫(-u^3 / √(1 - u^2)) du.
This integral can be solved using a u-substitution. Let v = 1 - u^2, then dv = -2u du. We can rewrite the integral as:
-1/2 ∫(u^2) * (1/√v) dv.
Step 7: Simplify the integrand once more.
The previous expression can be further simplified to:
-1/2 ∫((u^2) / √v) dv.
Step 8: Integrate the simplified expression.
The integral ∫((u^2) / √v) dv can be computed using the power rule. The result is:
-1/2 * (2/3) * v^(3/2) + C,
where C is the constant of integration.
Step 9: Substitute back the original variable.
Applying the substitution v = 1 - u^2, we obtain:
-1/2 * (2/3) * (1 - u^2)^(3/2) + C.
Step 10: Substitute back the original variable and simplify further.
Finally, substituting back u = cos(x) and simplifying, we arrive at the solution:
-(1/3) * (1 - cos^2(x))^(3/2) + C.
Therefore, the integral of sec^3(x) dx is given by:
-(1/3) * (1 - cos^2(x))^(3/2) + C.