Solve ∫ sec(t)(3sec(t)+7tan(t))

$$\int \sec ( t ) ( 3 \sec ( t ) + 7 \tan ( t ) )$$

Evaluate

$$\frac{x\left(7\sin(t)+3\right)}{\left(\cos(t)\right)^{2}}+С$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }t=\pi n_{1}+\frac{\pi }{2}$$

Solution Steps

Find the integral of \frac{3+7\sin(t)}{\cos(t)\cos(t)} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3+7\sin(t)}{\cos(t)\cos(t)}x$$

Simplify.

$$\frac{\left(3+7\sin(t)\right)x}{\left(\cos(t)\right)^{2}}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\begin{matrix}\frac{\left(3+7\sin(t)\right)x}{\left(\cos(t)\right)^{2}}+С_{3},&\end{matrix}$$

Differentiate w.r.t. x

$$\frac{7\sin(t)+3}{\left(\cos(t)\right)^{2}}$$