Question

Evaluate the integral $\int {\dfrac{{{{\sec }^2}x - 7}}{{{{\sin }^7}x}}} dx = $
$A)\left[ {\dfrac{{\tan x}}{{{{\sin }^7}x}}} \right] + c$
$B)\left[ {\dfrac{{\cos x}}{{{{\sin }^7}x}}} \right] + c$
$C)\left[ {\dfrac{{\sin x}}{{{{\cos }^7}x}}} \right] + c$
$D)\left[ {\dfrac{{\sin x}}{{{{\tan }^7}x}}} \right] + c$

Answer 1

Hint: The given question consists of two different trigonometric functions in the numerator and the denominator. Therefore, we need to find out the integral value through the Integration By parts method. Since it is constant, we will separate it from the attached trigonometric function. Then, we apply by parts for the expression which contains only trigonometric functions.

Complete answer:
Let,
$I = \int {\dfrac{{{{\sec }^2}x - 7}}{{{{\sin }^7}x}}} dx$
Now, we can separate the constant and write the expression as,
$I = \int {\dfrac{{{{\sec }^2}x}}{{{{\sin }^7}x}}} dx - \int {\dfrac{7}{{{{\sin }^7}x}}} dx$
We can write $\dfrac{1}{{\sin x}}$ as $\cos ecx$
Now, we have the expression,
$ \Rightarrow I = \int {\cos e{c^7}} x \times {\sec ^2}xdx - \int {7\cos e{c^7}} xdx$
We apply integration by parts for the above expression $\int {f\left( x \right)g\left( x \right)dx = f\left( x \right)\int {g\left( x \right)dx - \int {\left[ {\dfrac{d}{{dx}}f\left( x \right)\int {g\left( x \right)dx} } \right]dx} } } $, we get,
\[ \Rightarrow I = \cos e{c^7}x\int {{{\sec }^2}xdx} - \int {\left[ {\dfrac{d}{{dx}}\left[ {\cos e{c^7}x} \right]\int {{{\sec }^2}xdx} } \right]} - \int {7\cos e{c^7}} xdx\]
We know that \[\int {{{\sec }^2}xdx} = \tan x\] and $\dfrac{{d\left( {\cos ecx} \right)}}{{dx}} = - \cos ecx\cot x$.
\[ \Rightarrow I = \cos e{c^7}x\tan x - \int {\left[ {7\cos e{c^6}x\left( { - \cos ecx\cot x} \right)\tan x} \right]} - \int {7\cos e{c^7}} xdx\]
Now,
\[ \Rightarrow I = \cos e{c^7}x\tan x - \int {\left[ {7\cos e{c^6}x\left( { - \cos ecx\cot x} \right)\tan x} \right]} - \int {7\cos e{c^7}} xdx\]
Simplifying the expression, we get,
\[ \Rightarrow I = \cos e{c^7}x\tan x + \int {\left[ {7\cos e{c^7}x\cot x\tan x} \right]} - \int {7\cos e{c^7}} xdx\]
Now, we know that tangent and cotangent are reciprocal functions. So, we get,
\[ \Rightarrow I = \cos e{c^7}x\tan x + \int {7\cos e{c^7}xdx} - \int {7\cos e{c^7}} xdx\]
Now, cancelling the similar terms with opposite signs, we get,
Where, $\cos e{c^7}x$ can be expressed as $\dfrac{1}{{{{\sin }^7}x}}$
Therefore, the final answer will be
$I = \dfrac{{\tan x}}{{{{\sin }^7}x}} + c$
Hence, option (A) is the correct option.

Note:
The integration by parts method is useful when two functions are multiplied together and have no specific formula for simplification. If the terms are not in the multiplication form, we need to bring them down into that form using the basic formula. Only then we can use the formula and simplify.