Answer: [tex]2\sqrt{1+tant}+C[/tex]
Step-by-step explanation:
To integrate means to find the antiderivative of the function. For this problem, we can use u-substitution.
[tex]\int\limits {\frac{dt}{cos^2t\sqrt{1+tant} } } \[/tex]
Let's first use our identities to rewrite the function. Since [tex]\frac{1}{cosx} =secx[/tex], we can use this identity.
[tex]\int\limits {\frac{sec^2t}{\sqrt{1+tant} } } \,[/tex]
[tex]u=\sqrt{1+tant}[/tex]
[tex]du=\frac{sec^2t}{2\sqrt{1+tant} } dt[/tex]
Now that we have u and du, we can plug them back in.
[tex]\int\limits {2} \, du[/tex]
[tex]\int\limits{2} \, du=2u[/tex]
Since we know u, we can plug that in.
[tex]2\sqrt{1+tant}[/tex]
This may seem like the correct answer, but we forgot to add the constant.
[tex]2\sqrt{1+tant}+C[/tex]
