Evaluate the following ∫01x3dx(1+x2)3

Question

Evaluate the following  ∫01x3dx(1+x2)3

Caren · Accepted Answer

Step 1
Integrals and derivatives are considered as the inverses of each other. For example if we integrate a function and then differentiate the integrated function, we will obtain our original function back.
Integrals are generally used to calculate area under the curve and volume of solid whereas derivatives are used to test the continuity of function and find the critical points of function where the function attains its maximum and minimum values.
Step 2
The given integral is

2 \int_{0}^{1} \frac{x^{3} d x}{{(1 + x^{2})}^{3}}

.
Evaluating the integral using substitution method
Let

t = 1 + x^{2}

dt=2xdx

2 \int_{0}^{1} \frac{x^{3} d x}{{(1 + x^{2})}^{3}} = 2 \int_{1}^{2} \frac{(x^{2}) x d x}{(t^{3})}

= \int_{1}^{2} \frac{t - 1}{(t^{3})} d t

= \int_{1}^{2} (\frac{1}{t^{2}} - \frac{1}{t^{3}}) d t

= \int_{1}^{2} (t^{- 2} - t^{- 3}) d t

= {[\frac{t^{- 3 + 1}}{- 3 + 1} - \frac{t^{- 2 + 1}}{- 3 + 1}]}_{1}^{2}

= {[\frac{1}{2 t^{2}} - \frac{1}{t}]}_{1}^{2}

= \frac{1}{2 {(2)}^{2}}

= \frac{1}{8}

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Evaluate the following \int_{0}^{1}\frac{x^{3}dx}{(1+x^{2})^{3}}

Answered question

Answer & Explanation

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