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Chapter 7 Class 12 Integrals | 9 questions

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Question 1 of 9
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Question 1 of 9
The given integral \(\int f(x) d x\) can be transformed into another form by changing the independent variable \(x\) to \(t\) by substituting \(x=g(t)\)

Consider \(\quad \mathrm{I}=\int f(x) d x\)

Put \(x=g(t)\) so that \(\frac{d x}{d t}=g^{\prime}(t)\)

We write \(d x=g^{\prime}(t) d t\)

Thus

$$ \mathrm{I}=\int f(x) d x=\int f(g(t)) g^{\prime}(t) d t $$


This change of variable formula is one of the important tools available to us in the name of integration by substitution.
For example: \(\quad \int 2 x \sin \left(x^2+1\right) d x\)

Put

$$ \begin{aligned} & x^2+1=t \\ & 2 x d x=d t \end{aligned} $$


Thus,

$$ \begin{aligned} \int \sin (t) d t & =-\cos (t)+C \\ & =-\cos \left(x^2+1\right)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
\(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^2} d x\) is equal to:
Select one option. Answers are shown after the test.
Question 2 of 9
The given integral \(\int f(x) d x\) can be transformed into another form by changing the independent variable \(x\) to \(t\) by substituting \(x=g(t)\)

Consider \(\quad \mathrm{I}=\int f(x) d x\)

Put \(x=g(t)\) so that \(\frac{d x}{d t}=g^{\prime}(t)\)

We write \(d x=g^{\prime}(t) d t\)

Thus

$$ \mathrm{I}=\int f(x) d x=\int f(g(t)) g^{\prime}(t) d t $$


This change of variable formula is one of the important tools available to us in the name of integration by substitution.
For example: \(\quad \int 2 x \sin \left(x^2+1\right) d x\)

Put

$$ \begin{aligned} & x^2+1=t \\ & 2 x d x=d t \end{aligned} $$


Thus,

$$ \begin{aligned} \int \sin (t) d t & =-\cos (t)+C \\ & =-\cos \left(x^2+1\right)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
\(\int \tan x d x\) is equal to:
Select one option. Answers are shown after the test.
Question 3 of 9
The given integral \(\int f(x) d x\) can be transformed into another form by changing the independent variable \(x\) to \(t\) by substituting \(x=g(t)\)

Consider \(\quad \mathrm{I}=\int f(x) d x\)

Put \(x=g(t)\) so that \(\frac{d x}{d t}=g^{\prime}(t)\)

We write \(d x=g^{\prime}(t) d t\)

Thus

$$ \mathrm{I}=\int f(x) d x=\int f(g(t)) g^{\prime}(t) d t $$


This change of variable formula is one of the important tools available to us in the name of integration by substitution.
For example: \(\quad \int 2 x \sin \left(x^2+1\right) d x\)

Put

$$ \begin{aligned} & x^2+1=t \\ & 2 x d x=d t \end{aligned} $$


Thus,

$$ \begin{aligned} \int \sin (t) d t & =-\cos (t)+C \\ & =-\cos \left(x^2+1\right)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
\(\int \frac{2 x}{1+x^2} d x\) is equal to:
Select one option. Answers are shown after the test.
Question 4 of 9
The given integral \(\int f(x) d x\) can be transformed into another form by changing the independent variable \(x\) to \(t\) by substituting \(x=g(t)\)

Consider \(\quad \mathrm{I}=\int f(x) d x\)

Put \(x=g(t)\) so that \(\frac{d x}{d t}=g^{\prime}(t)\)

We write \(d x=g^{\prime}(t) d t\)

Thus

$$ \mathrm{I}=\int f(x) d x=\int f(g(t)) g^{\prime}(t) d t $$


This change of variable formula is one of the important tools available to us in the name of integration by substitution.
For example: \(\quad \int 2 x \sin \left(x^2+1\right) d x\)

Put

$$ \begin{aligned} & x^2+1=t \\ & 2 x d x=d t \end{aligned} $$


Thus,

$$ \begin{aligned} \int \sin (t) d t & =-\cos (t)+C \\ & =-\cos \left(x^2+1\right)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
\(\int \sin (a x+b) \cos (a x+b) d x\) is equal to:
Select one option. Answers are shown after the test.
Question 5 of 9
The given integral \(\int f(x) d x\) can be transformed into another form by changing the independent variable \(x\) to \(t\) by substituting \(x=g(t)\)

Consider \(\quad \mathrm{I}=\int f(x) d x\)

Put \(x=g(t)\) so that \(\frac{d x}{d t}=g^{\prime}(t)\)

We write \(d x=g^{\prime}(t) d t\)

Thus

$$ \mathrm{I}=\int f(x) d x=\int f(g(t)) g^{\prime}(t) d t $$


This change of variable formula is one of the important tools available to us in the name of integration by substitution.
For example: \(\quad \int 2 x \sin \left(x^2+1\right) d x\)

Put

$$ \begin{aligned} & x^2+1=t \\ & 2 x d x=d t \end{aligned} $$


Thus,

$$ \begin{aligned} \int \sin (t) d t & =-\cos (t)+C \\ & =-\cos \left(x^2+1\right)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
\(\int \frac{1}{x+x \log x} d x\) is equal to:
Select one option. Answers are shown after the test.
Question 6 of 9
$$ \begin{aligned} \int e^x\left[f(x)+f^{\prime}(x)\right] d x & \\ & =\int \boldsymbol{e}^x \boldsymbol{f}(\boldsymbol{x}) \boldsymbol{d} x+\int e^x f^{\prime}(x) d x \end{aligned} $$


Using integration by parts

$$ \begin{aligned} & =\boldsymbol{f}(\boldsymbol{x}) \int \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} \boldsymbol{x}-\int \boldsymbol{f}^{\prime}(\boldsymbol{x}) \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} x+\int f^{\prime}(x) e^x d x \\ & =f(x) e^x-\int f^{\prime}(x) e^x d x+\int f^{\prime}(x) e^x d x \\ & =e^x f(x)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
$$ \int e^x\left(\frac{x-1}{x^2}\right) d x= $$

\(\_\_\_\_\)
Select one option. Answers are shown after the test.
Question 7 of 9
$$ \begin{aligned} \int e^x\left[f(x)+f^{\prime}(x)\right] d x & \\ & =\int \boldsymbol{e}^x \boldsymbol{f}(\boldsymbol{x}) \boldsymbol{d} x+\int e^x f^{\prime}(x) d x \end{aligned} $$


Using integration by parts

$$ \begin{aligned} & =\boldsymbol{f}(\boldsymbol{x}) \int \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} \boldsymbol{x}-\int \boldsymbol{f}^{\prime}(\boldsymbol{x}) \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} x+\int f^{\prime}(x) e^x d x \\ & =f(x) e^x-\int f^{\prime}(x) e^x d x+\int f^{\prime}(x) e^x d x \\ & =e^x f(x)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
$$ \int e^x(1+x) d x= $$

\(\_\_\_\_\)
Select one option. Answers are shown after the test.
Question 8 of 9
$$ \begin{aligned} \int e^x\left[f(x)+f^{\prime}(x)\right] d x & \\ & =\int \boldsymbol{e}^x \boldsymbol{f}(\boldsymbol{x}) \boldsymbol{d} x+\int e^x f^{\prime}(x) d x \end{aligned} $$


Using integration by parts

$$ \begin{aligned} & =\boldsymbol{f}(\boldsymbol{x}) \int \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} \boldsymbol{x}-\int \boldsymbol{f}^{\prime}(\boldsymbol{x}) \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} x+\int f^{\prime}(x) e^x d x \\ & =f(x) e^x-\int f^{\prime}(x) e^x d x+\int f^{\prime}(x) e^x d x \\ & =e^x f(x)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
$$ \int_0^\pi e^x\left(\tan x+\sec ^2 x\right) d x= $$

\(\_\_\_\_\)
Select one option. Answers are shown after the test.
Question 9 of 9
$$ \begin{aligned} \int e^x\left[f(x)+f^{\prime}(x)\right] d x & \\ & =\int \boldsymbol{e}^x \boldsymbol{f}(\boldsymbol{x}) \boldsymbol{d} x+\int e^x f^{\prime}(x) d x \end{aligned} $$


Using integration by parts

$$ \begin{aligned} & =\boldsymbol{f}(\boldsymbol{x}) \int \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} \boldsymbol{x}-\int \boldsymbol{f}^{\prime}(\boldsymbol{x}) \boldsymbol{e}^{\boldsymbol{x}} \boldsymbol{d} x+\int f^{\prime}(x) e^x d x \\ & =f(x) e^x-\int f^{\prime}(x) e^x d x+\int f^{\prime}(x) e^x d x \\ & =e^x f(x)+C \end{aligned} $$


Based on the above information, answer any four of the following questions.
$$ \int \frac{x e^x}{(1+x)^2} d x= $$

\(\_\_\_\_\)
Select one option. Answers are shown after the test.