5.4 Indefinite Integrals and the Net Change Theorem/17

17. \begin{align}

&= \int_{}^{}1+tan^2 x\,dx \\[2ex] &= \int1+\frac{sin^2x}{cos^2x}\,dx \\[2ex] &= \int\frac{cos^2x+sin^2x}{cos^2x}\,dx \\[2ex] &= \cos^2x+sin^2x=1, thus &= \int\frac{1}{cos^2x}\,dx \\[2ex] &= \int\sec^2x\,dx \\[2ex] &= tanx+C\ \end{align}