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**Hint:**We know that antiderivative means integration. We need to find the integration of \[\dfrac{{\sin (2x)}}{{\cos (x)}}dx\]. Here we have an indefinite integral. In the numerator we have sine double angle, we know the sine double angle formula that is \[\sin (2x) = 2.\sin x.\cos x\]. We substitute this in the given problem and then we integrate with respect to ‘x’.

**Complete step-by-step solution:**

Given \[\int {\dfrac{{\sin (2x)}}{{\cos (x)}}dx} \].

We know \[\sin (2x) = 2.\sin x.\cos x\].

The term inside the integral symbol is called the integrand.

Then the integrand becomes

\[\dfrac{{\sin (2x)}}{{\cos (x)}} = \dfrac{{2.\sin x.\cos x}}{{\cos x}}\]

Cancelling the cosine function we have,

\[\dfrac{{\sin (2x)}}{{\cos (x)}} = 2.\sin x.\]

Now applying the integration we have

\[\int {\dfrac{{\sin (2x)}}{{\cos (x)}}dx} = \int {2.\sin x} .dx\]

\[ = \int {2.\sin x} .dx\]

Taking constant term outside the integral we have,

\[ = 2\int {\sin x} .dx\]

Integrating we have,

\[ = - 2\cos x + c\]

Thus we have

The antiderivative of \[\dfrac{{\sin (2x)}}{{\cos (x)}}dx\] is \[ - 2\cos x + c\]. Where ‘c’ is the integration constant.

**Note:**In the given above problem we have an indefinite integral, that is no upper and lower limit. Hence we add the integration constant ‘c’ after integrating. In a definite integral we will have an upper and lower limit, we don’t need to add integration constant in the case of definite integral. We have different integration rule:

The power rule: If we have a variable ‘x’ raised to a power ‘n’ then the integration is given by \[\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} \].

The constant coefficient rule: if we have an indefinite integral of \[K.f(x)\], where f(x) is some function and ‘K’ represent a constant then the integration is equal to the indefinite integral of f(x) multiplied by ‘K’. That is \[\int {K.f(x)dx = c\int {f(x)dx} } \].

The sum rule: if we have to integrate functions that are the sum of several terms, then we need to integrate each term in the sum separately. That is

\[\int {\left( {f(x) + g(x)} \right)dx = \int {f(x)dx} } + \int {g(x)dx} \]

For the difference rule we have to integrate each term in the integrand separately.

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