Name: Derivative of Logarithmic Functions
Uploaded: 2019-07-19
Duration: 12 min 13 s
Description: Let's go in and find out what is the derivative of functions containing logarithms and what is the formula for derivatives of logarithmic functions.

Question

Integrate the following with respect to $x$ .

$(1) x^{3}$

$(2) x - \frac{1}{x}$

$(3) e^{2 x} + \frac{1}{x^{2}}$

$(4) \frac{1}{2 x + 3}$

$(5) \cos (4 x + 3)$

$(6) \cos^{2} x$

Accepted Answer

$(1) y = x^{3}$

Integrating both sides with respect to $(2) x - \frac{1}{x}$ , we get,

$I = \int y d x = \int x^{3} d x = \frac{x^{4}}{4} + c [∵ \int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c]$

$(2) y = x - \frac{1}{x}$

Integrating both sides with respect to $x$ , we get,

$I = \int y d x = \int (x - \frac{1}{x}) d x [∵ \int \frac{1}{x} d x = \ln (x) + c]$

$I = \frac{x^{2}}{2} - \ln (x) + c$ .

$(2) x - \frac{1}{x}$

Integrating both sides with respect to $x$ , we get,

$I = \int y d x = \int (e^{2 x} + \frac{1}{x^{2}}) d x$

$I = \frac{e^{2 x}}{2} + \frac{x^{- 2 + 1}}{- 2 + 1} + c [∵ \int e^{a x} d x = \frac{e^{a x}}{a} + c]$

$I = \frac{e^{2 x}}{2} - \frac{1}{x} + c$ .

$(4) y = \frac{1}{2 x + 3}$

Integrating both sides with respect to $x$ , we get,

$I = \int y d x = \int \frac{d x}{2 x + 3} = \frac{\ln | 2 x + 3 |}{2} + c$ .

$(5) y = \cos (4 x + 3)$

Integrating both sides with respect to $x$ , we get,

$I = \int y d x = \int \cos (4 x + 3) d x = \frac{\sin (4 x + 3)}{4} + c$ .

$(6) y = \cos^{2} x$

Integrating both sides with respect to $x$ , we get,

$I = \int y d x$

$I = \int \cos^{2} x d x = \int (\frac{1 + \cos 2 x}{2}) d x$

$I = \int \frac{1}{2} d x + \int \frac{\cos 2 x}{2} d x$

$I = \frac{x}{2} + \frac{\sin 2 x}{2 \times 2} + c = \frac{1}{2} (x + \frac{\sin 2 x}{2}) + c$ .

Integrate the following with respect to $x$ .

$(1) x^{3}$

$(2) x - \frac{1}{x}$

$(3) e^{2 x} + \frac{1}{x^{2}}$

$(4) \frac{1}{2 x + 3}$

$(5) \cos (4 x + 3)$

$(6) \cos^{2} x$

Important Questions on Basic Mathematics

Learn and Prepare for any exam you want !

Integrate the following with respect to x. 1 x3 2 x-1x 3 e2x+1x2 4 12x+3 5 cos4x+3 6 cos2x

Important Questions on Basic Mathematics

Learn and Prepare for any exam you want !

Integrate the following with respect to $x$ .

$(1) x^{3}$

$(2) x - \frac{1}{x}$

$(3) e^{2 x} + \frac{1}{x^{2}}$

$(4) \frac{1}{2 x + 3}$

$(5) \cos (4 x + 3)$

$(6) \cos^{2} x$