Evaluate the integral. $$ \int_{0}^{\pi / 4} 10 \sec ^{2} x d x $$ $$ \int_{0}^{\pi / 4} 10 \sec ^{2} x d x=\square $$ (Type an exact answer.)

Question

UpStudy AI · Accepted Answer

We start by recalling the antiderivative of $$ \sec^2 x $$:

$$
\frac{d}{dx} \tan x = \sec^2 x.
$$

Thus, the antiderivative is:

$$
\int \sec^2 x \, dx = \tan x.
$$

Multiplying by 10, we have:

$$
\int 10 \sec^2 x \, dx = 10 \tan x.
$$

Now, evaluating the definite integral from $$ 0 $$ to $$ \frac{\pi}{4} $$:

$$
\int_{0}^{\pi/4} 10 \sec^2 x \, dx = 10 \tan x \Big|_{0}^{\pi/4} = 10 \left( \tan\frac{\pi}{4} - \tan 0 \right).
$$

We know that:

$$
\tan\frac{\pi}{4} = 1 \quad \text{and} \quad \tan 0 = 0.
$$

Substitute these values:

$$
10 \left( 1 - 0 \right) = 10.
$$

The exact answer is:

$$
\boxed{10}.
$$
The integral evaluates to 10.

Evaluate the integral. \( \int_{0}^{\pi / 4} 10 \sec ^{2} x d x \) \( \int_{0}^{\pi / 4} 10 \sec ^{2} x d x=\square \) (Type an exact answer.)