Find the indefinite integral. ∫(2x+1)ex2+xdx

Question

Find the indefinite integral.  ∫(2x+1)ex2+xdx

zurilomk4 · Accepted Answer

Step 1
Integration is summation of discrete data. The integral is calculated for the functions to find their area, displacement, volume, that occurs due to combination of small data.
Integration is of two types definite integral and indefinite integral. Indefinite integral are defined where upper and lower limits are not given, whereas in definite integral both upper and lower limit are there.
Step 2
The given integrand is

\int (2 x + 1) e^{x^{2} + x} d x

, consider the exponent

x^{2} + x

as u. Differentiate it with respect to x

u = x^{2} + x

\frac{d u}{d x} = \frac{d (x^{2} + x)}{d x}

= \frac{{d x}^{2}}{d x} + \frac{d x}{d x}

=2x+1
du=(2x+1)dx...(1)
Step 3
Substitute value of

x^{2} + x

as u and du=(2x+1)dx from equation (1) in

\int (2 x + 1) e^{x^{2} + x} d x

\int (2 x + 1) e^{x^{2} + x} d x = \int e^{u} d u

= e^{u} + C

= e^{x^{2}} + C

(As

x^{2} + x = u

)
Therefore, integration of

\int (2 x + 1) e^{x^{2} + x} d x

is equal to

e^{x^{2} + x} + C

.

Neil Dismukes · Accepted Answer

∫(2x+1)⋅ex2+xdx  We put the expression 2 * x + 1 under the differential sign, i.e.:  (2x+1)dx=d(x2+x),t=x2+x  Then the original integral can be written as follows:  ∫et⋅dt  This is a tabular integral:  ∫etdt=et+C  To write the final answer, it remains to substitute x2+x instead of t.  ex2+x+C

Vasquez · Accepted Answer

Given: ∫(2x+1)ex2+xdx Transform the expression ∫1dt Use ∫1dx=x to evaluate the integral t Substitute back t=ex2+x ex2+x Add C Answer: ex2+x+C

Find the indefinite integral. \int (2x+1)e^{x^{2}+x}dx

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