Find the indefinite integral of (1 + tan^2 y) with respect to y.

Step-by-step solution

1. 1

   Do not integrate term by term. Splitting into $\int 1\,dy+\int\tan^{2}y\,dy$ is legal but pointless: the second piece has no elementary antiderivative you can write down without first using the very identity that solves the whole problem.

2. 2

   Apply the Pythagorean identity. Starting from $\sin^2\theta+\cos^2\theta=1$ and dividing every term by $\cos^2\theta$ gives

   $\tan^{2}y+1=\sec^{2}y.$

   So the integrand is simply $\sec^{2}y$.

3. 3

   Recognise the derivative you already know. Since

   $\frac{d}{dy}\tan y=\sec^{2}y,$

   the integral is immediate:

   $\int\sec^{2}y\,dy=\tan y+C.$

4. 4

   State the result with its constant of integration.

   $\int\left(1+\tan^{2}y\right)dy=\tan y+C.$

5. 5

   Verify by differentiating back. Using the quotient rule on $\tan y=\dfrac{\sin y}{\cos y}$:

   $\frac{d}{dy}\tan y=\frac{\cos^{2}y+\sin^{2}y}{\cos^{2}y}=\frac{1}{\cos^{2}y}=\sec^{2}y=1+\tan^{2}y,$

   which is the original integrand exactly. Note the antiderivative is valid on each interval where $\cos y\neq 0$, that is between consecutive odd multiples of $\tfrac{\pi}{2}$.