Mathematics

# Solve $\int {x.\sqrt[5]{{x + 1}}dx}$

##### SOLUTION
$\int x.\sqrt[5]{x+1}dx...(1)$
put $x+1=t^{5}….(2)$
hence, $x=t^{5}- 1...(3)$  and $dx=5t^{4} dt...(4)$
substitute equation (2) (3) and (4) in equation (1) we get
$\int(t^{5}-1)(t)(5t^{4})dt$
$\int5(t^{5}-1)t^{5}dt$
$5\int(t^{10}-t^{5})dt$
$5\bigg[\dfrac{t^{11}}{11}-\dfrac{t^{6}}{6}\bigg]$
now, put value of $t=(x+1)^{\dfrac{1}{5}}$ from equation (2) in above equation we get.
$I=5\bigg[ \dfrac{(x+1)^{\dfrac{11}{5}}}{11}-\dfrac{(x+1)^{\dfrac{6}{5}}}{6}\bigg]+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
Prove that $\displaystyle\int^1_0x(1-x)^5dx=\dfrac{1}{42}$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The value of the integral $\int _{ 1 }^{ { 3 }^{ 1/n } }{ \cfrac { dx }{ x({ x }^{ n }+1) } }$ is
• A. $\cfrac { 1 }{ n } \log { \left( \cfrac { 2 }{ 3 } \right) }$
• B. $n\log { \left( \cfrac { 2 }{ 3 } \right) }$
• C. $n\log { \left( \cfrac { 3 }{ 2 } \right) }$
• D. $\cfrac { 1 }{ n } \log { \left( \cfrac { 3 }{ 2 } \right) }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following integral
$\int { \cfrac { \sin { 2x } }{ a\cos ^{ 2 }{ x } +b\sin ^{ 2 }{ x } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate : $\int {\dfrac{{{e^x}dx}}{{\left( {1 + {e^{2x}}} \right)}}}$.

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$