Use substitution to find the indefinite integral. ∫4x3e2x4dx

Question

Use substitution to find the indefinite integral.  ∫4x3e2x4dx

Mollie Nash · Accepted Answer

Step 1
Integration by substitution is a method to solve integrals line in the above question where the integral of product of two functions are to be determined and one function can be written as the derivative of the other.
Step 2
To determine the integral given by;
$\int 4 x^{3} e^{2 x^{4}} d x$
let ; $e^{2 x^{4}} = t$
$\frac{d (e^{2 x^{4}})}{d x} = \frac{d t}{d x}$
$4 \cdot 2 x^{3} \cdot e^{2 x^{4}} d x = d t$
$∴ 2 \cdot 4 x^{3} e^{2 x^{4}} d x = d t$
$4 x^{3} e^{2 x^{4}} d x = \frac{d t}{2}$
Therefore the integral becomes;
$\int 4 x^{3} e^{2 x^{4}} d x = \int \frac{d t}{2}$
$= \frac{1}{2} \int d t$
$= \frac{t}{2} + C$
$= \frac{e^{2 x^{4}}}{2} + C \dots [∵ t = e^{2 x^{4}}]$

amarantha41 · Accepted Answer

∫4x3⋅e2x4dx  The original integral can be simplified: We  ∫4x3⋅e2x4dx=∫x3⋅e2x4dx  bring the expression 4⋅x3 under the sign of the differential, i.e.:  Then the original integral can be written as follows:  ∫e2t4dt  This is a tabular integral:  ∫e2t4dt=e2t8+C  To to write down the final answer, it remains to substitute x4 instead of t.  e2x48+C

Vasquez · Accepted Answer

∫4x3e2x4dx4∗∫x3e2x4dx4∗∫et8dt4∗18∗∫etdtCalculate Integrate12et12e2x4e2x42Add Ce2x42+C

Use substitution to find the indefinite integral. \int 4x^{3}e^{2x^{4}}dx

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