Find a fraction that is equivalent to 1.5 over 9 with a whole number in both numerator and the denominator

Mathematics

1 answer:

Aneli [31]3 years ago

8 0

Answer: $\frac{1}{6}$

Step-by-step explanation:

We need to remember that equivalent fractions are defined as fractions whose numerators and denominator are different but both fractions represent the same value.

To find an equivalent fraction, we must multiply the numerator and the denominator by the same number (Which must be a nonzero whole number).

In this case, to find a fraction equivalent to $\frac{1.5}{9}$ , with a whole number in both numerator and the denominator, we can multiply the numerator and the denominator by 2. Then you get:

$\frac{1.5*2}{9*2}=\frac{3}{18}$

Reducing the fraction, you get:

$=\frac{1}{6}$

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A. Evaluate ∫20 tan 2x sec^2 2x dx using the substitution u = tan 2x.

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Answer:

The integral is equal to $5\sec^2(2x)+C$ for an arbitrary constant C.

Step-by-step explanation:

a) If $u=\tan(2x)$ then $du=2\sec^2(2x)dx$ so the integral becomes $\int 20\tan(2x)\sec^2(2x)dx=\int 10\tan(2x) (2\sec^2(2x))dx=\int 10udu=\frac{u^2}{2}+C=10(\int udu)=10(\frac{u^2}{2}+C)=5\tan^2(2x)+C$ . (the constant of integration is actually 5C, but this doesn't affect the result when taking derivatives, so we still denote it by C)

b) In this case $u=\sec(2x)$ hence $du=2\tan(2x)\sec(2x)dx$ . We rewrite the integral as $\int 20\tan(2x)\sec^2(2x)dx=\int 10\sec(2x) (2\tan(2x)\sec(2x))dx=\int 10udu=5\frac{u^2}{2}+C=5\sec^2(2x)+C$ .

c) We use the trigonometric identity $\tan(2x)^2+1=\sec(2x)^2$ is part b). The value of the integral is $5\sec^2(2x)+C=5(\tan^2(2x)+1)+C=5\tan^2(2x)+5+C=5\tan^2(2x)+C$ . which coincides with part a)

Note that we just replaced 5+C by C. This is because we are asked for an indefinite integral. Each value of C defines a unique antiderivative, but we are not interested in specific values of C as this integral is the family of all antiderivatives. Part a) and b) don't coincide for specific values of C (they would if we were working with a definite integral), but they do represent the same family of functions.

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