The value of for which 4 x^3 + 4^x 4^x + x^4 , ,dx = ( 4^x + x^4 … - Vidyadip

MCQ

The value of $\lambda$ for which $\int {\frac{{4{x^3} + \lambda {4^x}}}{{{4^x} + {x^4}}}} \,\,dx = \log ({4^x} + {x^4}) + c$ is

A
$1$
✓
$log_e4$
C
$log_4e$
D
$4$

✓

Answer

Correct option: B.

$log_e4$

b
Put $4^{x}+x^{4}=t$

$\left(4^{x} \ln 4+4 x^{3}\right) d x=d t$

$\int {\frac{{{\rm{dt}}}}{{\rm{t}}}} = \ln {\rm{t}} + {\rm{c}}$

$ = \ln \left| {{4^x} + {{\rm{x}}^4}} \right| + {\rm{c}}$

$\therefore \lambda=\ln 4$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 7.1 inderfinite integral→7.1 inderfinite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $\ell_1$ and $\ell_2$ be the lines $\overrightarrow{\mathfrak{1}}_1=\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $\vec{r}_2=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}+\hat{\mathrm{k}})$, respectively. Let $\mathrm{X}$ be the set of all the planes $\mathrm{H}$ that contain the line $\ell_1$. For a plane $\mathrm{H}$, let $\mathrm{d}(\mathrm{H})$ denote the smallest possible distance between the points of $\ell_2$ and $H$. Let $\mathrm{H}_0$ be plane in $X$ for which $\mathrm{d}\left(\mathrm{H}_0\right)$ is the maximum value of $\mathrm{d}(\mathrm{H})$ as $\mathrm{H}$ varies over all planes in $\mathrm{X}$.

Match each entry in List-$I$ to the correct entries in List-$II$.

List-$I$	List-$II$
($P$) The value of $\mathrm{d}\left(\mathrm{H}_0\right)$ is	($1$) $\sqrt{3}$
($Q$) The distance of the point $(0,1,2)$ from $\mathrm{H}_0$ is	($2$) $\frac{1}{\sqrt{3}}$
($R$) The distance of origin from $\mathrm{H}_0$ is	($3$) $0$
($S$) The distance of origin from the point of intersection of planes $\mathrm{y}=\mathrm{z}, \mathrm{x}=1$ and $\mathrm{H}_0$ is	($4$) $\sqrt{2}$
	($5$) $\frac{1}{\sqrt{2}}$

The corret option is :

The direction ratios of the vector $3 \vec{a}+2 \vec{b}$, where $\vec{a}=\hat{i}+\hat{j}-2 \hat{k}$ and $\vec{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}$ are

Let $f, g: R \to R$ be two functions defined by $f(x)\, = \,\left\{ {\begin{array}{*{20}{c}}
{x\,\sin \,\left( {\frac{1}{x}} \right),\,x\, \ne \,0\,\,\,\,\,\,\,\,\,\,}\\
{0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x\, = 0\,\,\,\,\,\,\,\,\,}
\end{array}} \right.,$ and $g(x) =x\,f(x)$

Statement $I:$ $f$ is a continuous function at $x = 0.$
Statement $II:$ $g$ is a differentiable function at $x = 0.$

The absolute value of $\int\limits_{10}^{19} {\,\,\frac{{\sin \,x}}{{1\,\, + \,\,{x^8}}}} $ is less than :

The function $f$ satisfies the functional equation $3f(x) + 2f\left( {\frac{{x + 59}}{{x - 1}}} \right) = 10x + 30$ for all real $x \ne 1$. The value of $f(7)$ is

If ${x^{2/3}} + {y^{2/3}} = {a^{2/3}}$, then ${{dy} \over {dx}} = $

In $( - 4,\,4)$ the function $f(x) = \int\limits_{ - 10}^x {({t^4} - 4){e^{ - 4t}}dt} $ has

Let × be the binary operation on N given by a × b = HCF (a, b) where, a, b ∈ N. Find the value of 22 × 4.

1
2
3
4

$\int_{}^{} {\sec x{{\tan }^3}x\;dx = } $

The value of $\int {{e^{2x}}(2\sin 3x + 3\cos 3x)\,\,dx} $ is