Question:

If $I =\int\frac{x^{5}}{\sqrt{1+x^{3}}}dx$ , then I is equal to

Updated On: Apr 21, 2023
  • $\frac{2}{9}\left(1+x^{3}\right)^{\frac{5}{2}}+\frac{2}{3}\left(1+x^{3}\right)^{\frac{3}{2}}+C$
  • $log\left|\sqrt{x}+\sqrt{1+x^{3}}\right|+C$
  • $log\left|\sqrt{x}-\sqrt{1+x^{3}}\right|+C$
  • $\frac{2}{9}\left(1+x^{3}\right)^{\frac{3}{2}}-\frac{2}{3}\left(1+x^{3}\right)^{\frac{1}{2}}+C$
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The Correct Option is D

Solution and Explanation

$I =\int\frac{x^{5}}{\sqrt{1+x^{3}}}dx=\int \frac{x^{3}.x^{2}}{\sqrt{1^{.}+x^{3}}}dx$
$Let 1 + x^{3} = t^{2}, so that 3x^{2} dx = 2t dt$
$\Rightarrow x^{2} dx = \frac{2}{3}td t$
$\therefore I =\int\frac{\left(t^{2}-1\right)\frac{2}{3}t dt}{t}=\frac{2}{3}\int t^{2} -1)dt$
$=\frac{2}{3}\left(\frac{t^{3}}{3}-t\right)+C =\frac{2}{3}\left[\frac{\left(1+x^{3}\right)^{3/2}}{3}-\left(1+x^{3}\right)^{\frac{1}{2}}\right]+C$
$=\frac{2}{9}\left(1+x^{3}\right)^{3/2}-\frac{2}{3}\left(1+x^{3}\right)^{1/2}+C$
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Top Questions on Methods of Integration

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Concepts Used:

Methods of Integration

Given below is the list of the different methods of integration that are useful in simplifying integration problems:

Integration by Parts:

 If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:

∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C

Here f(x) is the first function and g(x) is the second function.

Method of Integration Using Partial Fractions:

The formula to integrate rational functions of the form f(x)/g(x) is:

∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx

where

f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and

g(x) = q(x).s(x)

Integration by Substitution Method

Hence the formula for integration using the substitution method becomes:

∫g(f(x)) dx = ∫g(u)/h(u) du

Integration by Decomposition

Reverse Chain Rule

This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,

∫g'(f(x)) f'(x) dx = g(f(x)) + C

Integration Using Trigonometric Identities