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JAMB Mathematics 2012 Paper

Question : 31

Total: 48

Evaluate

\int\limits_{0\sec^2}^{\frac{\pi}{4}} \theta \, d\theta

Solution:

Concept:

Use the standard integral

\int \sec^2\theta\,d\theta = \tan\theta + C

and then apply the limits of integration.

Explanation:

We need to evaluate

\displaystyle \int_{0}^{\frac{\pi}{4}} \sec^2\theta \, d\theta

No substitution is needed because the integrand is the derivative of

\tan\theta

So, the antiderivative is

\tan\theta

By the fundamental theorem of calculus:

\displaystyle \int_{0}^{\frac{\pi}{4}} \sec^2\theta \, d\theta = \left[\tan\theta\right]_{0}^{\frac{\pi}{4}}

At the upper limit:

\tan\frac{\pi}{4} = 1

At the lower limit:

\tan 0 = 0

Therefore, the value is

1 - 0 = 1

Answer:

1

, which is option A.

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