Question

How do you find the antiderivative of $\int {\dfrac{{\sin x}}{{{{\cos }^3}x}}dx} ?$

Answer 1

In order to solve this integral first we will assume $\cos x = t$ and differentiate it and transfer the given integral in terms of $t$ . And then we will use the formula as, $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c}$ to solve the given integral. And finally, we will substitute the value of $t$ to get the required result.

Complete step by step answer:
We have to find the antiderivative of $\int {\dfrac{{\sin x}}{{{{\cos }^3}x}}dx}$
Let us consider the given integral as,
$I = \int {\dfrac{{\sin x}}{{{{\cos }^3}x}}dx} {\text{ }} - - - \left( i \right)$
Now let us assume
$\cos x = t{\text{ }} - - - \left( {ii} \right)$
As we know that
$\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x$
So, by differentiating both the sides of equation $\left( {ii} \right)$ w.r.t $x$ we get
$- \sin x = \dfrac{{dt}}{{dx}}$
On multiplying by $dx$ both sides, we get
$- \sin xdx = dt{\text{ }}$
On multiplying with negative sign both the sides, we get
$\Rightarrow \sin xdx = - dt{\text{ }} - - - \left( {iii} \right)$
Now substituting the value from equation $\left( {ii} \right)$ and equation $\left( {iii} \right)$ in equation $\left( i \right)$ we get
$I = \int {\dfrac{{ - dt}}{{{t^3}}}}$
Now we know that
$\dfrac{1}{{{a^n}}} = {a^{ - n}}$
Therefore, we get
$I = \int { - {t^{ - 3}}dt}$
As we know that
$\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c}$
Therefore, we get
$I = \dfrac{{ - {t^{ - 3 + 1}}}}{{ - 3 + 1}} + c$
On solving the numerator and denominator, we get
$I = \dfrac{{ - {t^{ - 2}}}}{{ - 2}} + c$
On cancelling the negative sign, we get
$I = \dfrac{{{t^{ - 2}}}}{2} + c$
We know that
${a^{ - n}} = \dfrac{1}{{{a^n}}}$
Therefore, from the above equation, we get
$I = \dfrac{1}{{2{t^2}}} + c$
Now using equation $\left( {ii} \right)$ substitute the value of $t$
Therefore, we get
$I = \dfrac{1}{{2{{\cos }^2}x}} + c$
Hence, the antiderivative of $\int {\dfrac{{\sin x}}{{{{\cos }^3}x}}dx}$ is $\dfrac{1}{{2{{\cos }^2}x}} + c$

Note:
Antiderivative is another name of the inverse derivative, or the indefinite integral. Always remember while calculating antiderivatives never forget to add constant $c$ in the final result. But students should know that constant will be used only if we have indefinite integral.