Find the trigonometric integral
(a) ∫▒tan〖θ 〖sec〗^4 〗 θ dθ
is that tan(θ) * sec^4(θ) or what?
(sec) by itself is meaningless.
To find the integral ∫ tan(θ) sec^4(θ) dθ, we can use integration by parts or substitution.
Let's try using the method of substitution:
Step 1: Start by identifying a suitable substitution. In this case, we can let u = tan(θ) and du = sec^2(θ) dθ.
Step 2: Rewrite the integral in terms of the new variable. Substituting u and du, we have ∫ u sec^2(θ) dθ.
Step 3: Convert sec^2(θ) to terms of u. Recall that sec^2(θ) = 1 + tan^2(θ). Therefore, we can rewrite sec^2(θ) as 1 + u^2.
Step 4: Substitute the new expression for sec^2(θ) in the integral. The integral becomes ∫ u(1 + u^2) du.
Step 5: Expand the expression inside the integral. Multiplying the terms out, we have ∫ u + u^3 du.
Step 6: Integrate each term separately. The integral of u is (1/2)u^2, and the integral of u^3 is (1/4)u^4.
Step 7: Combine the integrals. The antiderivative of u + u^3 is (1/2)u^2 + (1/4)u^4.
Step 8: Replace u with the original variable θ. Remember that u = tan(θ). Therefore, the final antiderivative is (1/2)tan^2(θ) + (1/4)tan^4(θ) + C, where C is the constant of integration.
So, the trigonometric integral ∫ tan(θ) sec^4(θ) dθ is equal to (1/2)tan^2(θ) + (1/4)tan^4(θ) + C.