Mathematics

# If $\displaystyle \int { \left( u\cfrac { dv }{ dx } \right) } dx=uv-\int { wdx }$, then $w=$

$v\cfrac { du }{ dx }$

##### SOLUTION
Given,

$\displaystyle \int \left ( u\dfrac{dv}{dx} \right )dx=uv-\int wdx$

it is called as, product rule integration,
$\displaystyle \int u.v \ dx=u\int v\ dx$ $\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx$

$\displaystyle \int \left ( u\dfrac{dv}{dx} \right )dx$

$\displaystyle =uv-\int v\dfrac{du}{dx}dx$

$\displaystyle \therefore w=v\dfrac{du}{dx}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Subjective Medium
Evaluate  :  $I = \int {\dfrac{{x + 4}}{{{x^2} + 5}}} dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
If $\displaystyle \int \frac{3e^{x}-5e^{-x}}{4e^{x}+5e^{-x}}dx=ax+b \ln (4e^{x}+5e^{-x})+C$, then
• A. $\displaystyle a=\frac{1}{8}, b=\frac{7}{8}$
• B. $\displaystyle a=-\frac{1}{8}, b=-\frac{7}{8}$
• C. $\displaystyle a=\frac{1}{8}, b=-\frac{7}{8}$
• D. $\displaystyle a=-\frac{1}{8}, b=\frac{7}{8}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following integrals.
$\int {\frac{{dx}}{{\sqrt {2x - 3{x^2} + 1} }}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:
$\displaystyle\int^{\pi/2}_0\dfrac{\sin x\cos x}{(1+\sin^4x)}dx$.

Consider the integrals $I_1=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{tan x}}$ and $I_2=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{sin x}dx}{\sqrt{sin }x+\sqrt{cos}x}$