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

Pretoto4o

Pretoto4o

Answered question

2021-12-05

Evaluate each integral.
04sech2xxdx

Answer & Explanation

Opeance1951

Opeance1951

Beginner2021-12-06Added 26 answers

Step 1
We have to find definite integral:
04sech2xxdx
This will be solved by substitution method or change of variable.
Assuming t=x
Limit will change as following,
x0t0
x4t2
Differentiating with respect to x the assumed value, we get
dtdx=dxdx
dtdx=12x
2dt=dxx
Step 2
Substituting above value in the definite integral, we get
04sech2xxdx=02sech2t(2dt)
=202sech2tdt
=2[tanht]02 (since sech2xdx=tanhx+C)
=2[tanh2tanh0]
=2[tanh20]
=2tanh2
Hence, value of definite integral is 2tanh(2).

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