Mathematics

# Solve :$\displaystyle \int x \sqrt{x-1 dx}$

##### SOLUTION
$\displaystyle \int x\sqrt {x-1}dx$
By barts
$=x\dfrac {2}{3}(x-1)^{3/2}-\displaystyle \int \dfrac {2}{3}(x-1)^{3/2} dx$
$=\dfrac {2x}{3}(x-1)^{3/2}-\dfrac {2}{3} \dfrac {(x-1)^{3/2 +1}}{\dfrac {3}{2}+1}+C$
$\dfrac {2x(4+1)^{3/2}}{3}-\dfrac {4}{15}(x-1)^{5/2}+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
If $I= \displaystyle \int_{0}^{\pi }\left ( \pi x-x^{2} \right )^{100}\sin 2x\: dx$, then value of $I$ is?
• A. $\pi ^{100}$
• B. $\displaystyle \frac{1}{2}\left ( \pi ^{100}-\pi ^{97} \right )$
• C. $\displaystyle \frac{1}{2}\left ( \pi ^{100}+\pi ^{97} \right )$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Evaluate : $\displaystyle \int\frac {\sec\sqrt{x}.\tan\sqrt{x}}{\sqrt{x}}dx$
• A. $\sec\sqrt{x}+c$
• B. $\tan\sqrt{x}+c$
• C. $\displaystyle \frac{1}{2}\sec\sqrt{x}+c$
• D. $2\sec\sqrt{x}+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate the given integral.
$\int { { e }^{ x } } \left( \tan { x } -\log { \cos { x } } \right) dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
solve the following.
$\int\limits_0^1 {\frac{{\log (1 + t)}}{{(1 + {t})^2}}} dt$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard

In calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$ $\int\, u^{n}(x)v_{n}(x)\, dx$ where $v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration  by parts is especially useful when calculating $\int P_{n}(x)\, Q(x)\, dx$, where $P_{n}(x)$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.