Question

Evaluate: $\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{\sec x}}{ \sqrt[3]{\sec x}+\sqrt[3]{cosec\ x}}} \, dx$

Answer 1

Answer:

$\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{\sec x}}{ \sqrt[3]{\sec x}+\sqrt[3]{cosec\ x}}} \, dx=\pi/4$

Step-by-step explanation:

$Let\:I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{\sec x}}{ \sqrt[3]{\sec x}+\sqrt[3]{cosec\ x}}} \, dx$........(1)

Using, the property

$\boxed{\int\limits^{a}_0 {f(x)}\:dx=\int\limits^{a}_0 {f(a-x)}\:dx}$

$I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{\sec(\frac{\pi}{2}-x)}}{ \sqrt[3]{\sec(\frac{\pi}{2}-x)}+\sqrt[3]{cosec(\frac{\pi}{2}-x)}}} \, dx$

$I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{cosecx}}{ \sqrt[3]{cosecx}+\sqrt[3]{secx}}} \, dx$

$Let\:I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{cosec x}}{ \sqrt[3]{sec x}+\sqrt[3]{cosecx}}} \, dx$........(2)

Adding (1) and (2), we get

$2I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{secx}}{ \sqrt[3]{secx}+\sqrt[3]{cosecx}}}\,dx+\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{cosec x}}{ \sqrt[3]{secx}+\sqrt[3]{cosecx}}} \, dx$

$2I=\int\limits^{\pi/2}_0 {\frac{\sqrt[3]{sec x}+\sqrt[3]{cosecx}}{ \sqrt[3]{\sec x}+\sqrt[3]{cosec\ x}}}\,dx$

$2I=\int\limits^{\pi/2}_0 {1}\:dx$

$2I=[x]^{\pi/2}_0$

$2I=\pi/2-0$

$2I=\frac{\pi}{2}$

$\implies\:I=\frac{\pi}{4}$

Answer 2

Answer:

$\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{\sec x}}{ \sqrt[3]{\sec x}+\sqrt[3]{cosec\ x}}}dx=\frac{\pi}{4}$

Step-by-step explanation:

In the question,

We have to integrate the term,

Let us say,

$I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{\sec x}}{ \sqrt[3]{\sec x}+\sqrt[3]{cosec\ x}}}dx$

Now, we know from one of the properties of integration,

$\int\limits^{a}_0 {f(x)}\ dx=\int\limits^{a}_0 {f(a-x)dx$

Now using the property in the given equation we get,

$I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{\sec(\frac{\pi}{2}-x)}}{ \sqrt[3]{\sec(\frac{\pi}{2}-x)}+\sqrt[3]{cosec(\frac{\pi}{2}-x)}}}dx\\I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{cosecx}}{ \sqrt[3]{cosecx}+\sqrt[3]{secx}}}dx\\So,\\I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{cosec x}}{ \sqrt[3]{sec x}+\sqrt[3]{cosecx}}}dx$

Now, on adding this final equation with the given equation we get,

$2I=\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{secx}}{ \sqrt[3]{secx}+\sqrt[3]{cosecx}}}dx+\int\limits^{\pi/2}_0 {\frac{ \sqrt[3]{cosec x}}{ \sqrt[3]{secx}+\sqrt[3]{cosecx}}}dx\\2I=\int\limits^{\pi/2}_0 {\frac{\sqrt[3]{sec x}+\sqrt[3]{cosecx}}{ \sqrt[3]{\sec x}+\sqrt[3]{cosec\ x}}}dx\\2I=\int\limits^{\pi/2}_0 {1}dx\\So,\\2I=[x]^{\pi/2}_0\\2I=\pi/2-0\\So,\\I=\frac{\pi}{4}$

Therefore, on integrating the given equation we get,

$I=\frac{\pi}{4}$

Therefore, the evaluated solution of the equation is given by,

$I=\frac{\pi}{4}$