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Chapter 5: Problem 10

Symmetry in integrals Use symmetry to evaluate the following integrals. $$\int_{-\pi / 4}^{\pi / 4} \cos x d x$$

Short Answer

Expert verified
$$ \int_{-\pi / 4}^{\pi / 4} \cos x d x $$ Answer: The value of the definite integral is 鈭�2.

Step by step solution

01

Identify the symmetry of the function

Observe the given integral is over the interval \([-\pi / 4, \pi / 4]\). The cosine function has an even symmetry about the y-axis, which means \(\cos (-x) = \cos x\). Therefore, the function of interest, \(\cos x\), is an even function.
02

Use the symmetry property to evaluate the integral

Since the function is even and the integration interval is symmetric about the y-axis, we can rewrite the given integral as: $$ \int_{-\pi / 4}^{\pi / 4} \cos x d x = 2 \int_{0}^{\pi / 4} \cos x d x $$ We just need to multiply the integral over half the interval by 2, since symmetric intervals will have equal areas.
03

Evaluate the simplified integral

Now, we can find the antiderivative of \(\cos x\). We know that: $$ \int \cos x dx = \sin x + C $$ Since this is a definite integral, we don't need to worry about the constant of integration \(C\). We can now plug the limits of integration into the antiderivative to find the value of the definite integral: $$ 2\left(\sin\frac{\pi}{4} - \sin 0 \right) = 2\left(\frac{\sqrt{2}}{2} - 0\right) = \sqrt{2} $$ So, the value of the given integral is: $$ \int_{-\pi / 4}^{\pi / 4} \cos x d x = \sqrt{2} $$

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Most popular questions from this chapter

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{\sqrt{2}}^{2} \frac{d x}{x \sqrt{x^{2}-1}}$$

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