Mathematics

# Evaluate: $\displaystyle\int \dfrac{ dx} {{{\cos }^2}x{{\left( {1 - \tan x} \right)}^2}}$

##### SOLUTION
$I=\int \dfrac{1}{\cos^{2} x (1- \tan x)^2} dx$

$=\int \dfrac{\sec^{2}x}{(1-\tan x)^{2}} dx$

Let $\tan x=t \Rightarrow \sec^{2} xdx=dt$

$I=\int \dfrac{dt}{(1-t)^{2}}$

$I=\dfrac{1}{(1-t)}+C$

$I=\dfrac{1}{(1-\tan x)}+C$

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

#### Realted Questions

Q1 Multiple Correct Hard
If $\displaystyle \int \frac {xe^x}{\sqrt {1 + e^x}} dx = f(x) \sqrt {1 + e^x} -2 \log \: g(x) + C$, then
• A. $\displaystyle f(x) = x - 1$
• B. $\displaystyle g(x) = \frac {\sqrt {1 + e^x} + 1}{\sqrt {1 + e^x} - 1}$
• C. $\displaystyle g(x) = \frac {\sqrt {1 + e^x} - 1}{\sqrt {1 + e^x} + 1}$
• D. $\displaystyle f(x) = 2(x - 2)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following integral:
$\displaystyle\int^a_0\dfrac{x}{\sqrt{a^2+x^2}}dx$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate the following function with respect to $x$
$\dfrac{(\sin^{-1}{x})^{3/2}}{\sqrt{1-x^{2}}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
$\displaystyle\int \dfrac{7x-4}{(x-1)^2(x+2)}dx$.

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.