Evaluate each integral. ∫04sec⁡h2xxdx

Question

Evaluate each integral.  ∫04sec⁡h2xxdx

Opeance1951 · Accepted Answer

Step 1
We have to find definite integral:
$\int_{0}^{4} \frac{{\sec h}^{2} \sqrt{x}}{\sqrt{x}} d x$
This will be solved by substitution method or change of variable.
Assuming $t = \sqrt{x}$
Limit will change as following,
$x \Rightarrow 0 \Rightarrow t \Rightarrow 0$
$x \Rightarrow 4 \Rightarrow t \Rightarrow 2$
Differentiating with respect to x the assumed value, we get
$\frac{d t}{d x} = \frac{d \sqrt{x}}{d x}$
$\frac{d t}{d x} = \frac{1}{2 \sqrt{x}}$
$2 d t = \frac{d x}{\sqrt{x}}$
Step 2
Substituting above value in the definite integral, we get
$\int_{0}^{4} \frac{{\sec h}^{2} \sqrt{x}}{\sqrt{x}} d x = \int_{0}^{2} {\sec h}^{2} t (2 d t)$
$= 2 \int_{0}^{2} \sec h^{2} t d t$
$= 2 {[\tan h t]}_{0}^{2}$ (since $\int {\sec h}^{2} x d x = \tan h x + C$ )
$= 2 [\tan h 2 - \tan h 0]$
$= 2 [\tan h 2 - 0]$
$= 2 \tan h 2$
Hence, value of definite integral is $2 \tan h (2)$ .

Evaluate each integral. \int_{0}^{4}\frac{\sec h^{2}\sqrt{x}}{\

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