Evaluate the indefinite integral. ∫sec2(x)dxx

Question

Evaluate the indefinite integral.  ∫sec2(x)dxx

vicki331g8 · Accepted Answer

Step 1
We have to evaluate the integral:

\int \frac{\sec^{2} (\sqrt{x}) d x}{\sqrt{x}}

We will use substitution method since derivatives of one function is present in the integral.
So assuming,

t = \sqrt{x}

differentiating with respect to x,

\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}}

Substituting above values, we get

\int \frac{\sec^{2} (\sqrt{x}) d x}{\sqrt{x}} = \int \sec^{2} (t) (2 d t)

= 2 \int \sec^{2} (t) d t

Step 2
Since we know that

\int \sec^{2} x d x = \tan x + C

therefore,

2 \int \sec^{2} (t) d t = 2 \tan (t) + C

= 2 \tan (\sqrt{x}) + C

Where, C is an arbitrary constant.
Hence, value of given integral is

2 \tan (\sqrt{x}) + C

.

usumbiix · Accepted Answer

∫sec2(x)xdx  Substitution u=x⇒dudx=12x⇒dx=2xdu  =2∫sec2(u)du  ∫sec2(u)du  This is the well-known tabular integral:  =tan⁡(u)  We substitute the already calculated integrals:  2∫sec2(u)du  =2tan⁡(u)  Reverse replacement u=x:  =2tan⁡(x)  Solution:  =2tan⁡(x)+C

karton · Accepted Answer

∫sec⁡(x)2xdx ∫2sec⁡(t)2dt 2×∫sec⁡(t)2dt 2tan⁡(t) 2tan⁡(x) Add C∈R Solution: 2tan⁡(x)+C,C∈R

Evaluate the indefinite integral. \int \frac{\sec^{2}(\sqrt{x})dx}{\sqrt{x}}

Answered question

Answer & Explanation

New Questions in Calculus 2