Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

int_(0)^(sqrt3)(xdx)/(sqrt(x^(4)+16))

∫03xdxx4+16 \int_{0}^{\sqrt{3}} \frac{x d x}{\sqrt{x^{4}+16}}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Calculusright chevron icon
Evaluate definite integrals using the chain ruleright chevron icon

Full solution

Q. ∫03xdxx4+16 \int_{0}^{\sqrt{3}} \frac{x d x}{\sqrt{x^{4}+16}}
  1. Set up integral: Set up the integral for evaluation.\newlineWe need to evaluate the integral of the function x dxx4+16\frac{x \, dx}{\sqrt{x^4 + 16}} from 00 to 3\sqrt{3}.
  2. Find substitution: Look for a substitution that simplifies the integral.\newlineNotice that the denominator contains an expression of the form x4+16x^4 + 16, which suggests a trigonometric substitution. However, since the x4x^4 term complicates a direct trigonometric substitution, we might consider a substitution that simplifies the x4x^4 term. Let's try the substitution u=x2u = x^2, which implies that du=2xdxdu = 2x dx.
  3. Rewrite in terms of uu: Rewrite the integral in terms of uu.\newlineSubstituting u=x2u = x^2 and du=2xdxdu = 2x dx into the integral, we get:\newline∫03xdxx4+16=12∫03duu2+16\int_{0}^{\sqrt{3}} \frac{x dx}{\sqrt{x^4 + 16}} = \frac{1}{2} \int_{0}^{3} \frac{du}{\sqrt{u^2 + 16}}\newlineWe divided by 22 because we replaced 2xdx2x dx with dudu.
  4. Evaluate with new variable: Evaluate the integral with the new variable uu. Now we have the integral 12∫03duu2+16\frac{1}{2} \int_{0}^{3} \frac{du}{\sqrt{u^2 + 16}}, which is a standard form that can be solved using a trigonometric substitution or by recognizing it as a form of the inverse hyperbolic sine function, asinh(u4)\text{asinh}(\frac{u}{4}). The derivative of asinh(u4)\text{asinh}(\frac{u}{4}) is 1u2+16\frac{1}{\sqrt{u^2 + 16}}, which matches our integrand.
  5. Find antiderivative: Find the antiderivative.\newlineThe antiderivative of 1u2+16\frac{1}{\sqrt{u^2 + 16}} is asinh(u4)\text{asinh}(\frac{u}{4}), so the antiderivative of our integral is:\newline12asinh(u4)+C\frac{1}{2} \text{asinh}(\frac{u}{4}) + C
  6. Substitute and evaluate: Substitute back in terms of xx and evaluate the definite integral.\newlineSubstituting back u=x2u = x^2, we have:\newline12\frac{1}{2} asinh(x24)\left(\frac{x^2}{4}\right) + CC\newlineNow we can evaluate the definite integral from 00 to 3\sqrt{3}:\newline12\frac{1}{2} [asinh((3)24)\left(\frac{(\sqrt{3})^2}{4}\right) - asinh(024)\left(\frac{0^2}{4}\right)]\newline= 12\frac{1}{2} [asinhu=x2u = x^211 - asinh00]
  7. Calculate inverse hyperbolic sine: Calculate the values of the inverse hyperbolic sine function.\newlineasinh(34)\text{asinh}(\frac{3}{4}) can be found using a calculator or a table of values, and asinh(0)\text{asinh}(0) is 00.\newlineSo the definite integral is:\newline12[asinh(34)−0]\frac{1}{2} [\text{asinh}(\frac{3}{4}) - 0]\newline= 12asinh(34)\frac{1}{2} \text{asinh}(\frac{3}{4})
  8. Provide final answer: Provide the final answer.\newlineThe evaluated integral from 00 to 3\sqrt{3} of (x dx)/(x4+16)(x \, dx) / (\sqrt{x^4 + 16}) is 12asinh(34)\frac{1}{2} \text{asinh}(\frac{3}{4}).

More problems from Evaluate definite integrals using the chain rule

Question
Consider the following problem:\newlineThe water level under a bridge is changing at a rate of r(t)=40sin⁡(πt6) r(t)=40 \sin \left(\frac{\pi t}{6}\right) centimeters per hour (where t t is the time in hours). At time t=3 t=3 , the water level is 9191 centimeters. By how much does the water level change during the 4th  4^{\text {th }} hour?\newlineWhich expression can we use to solve the problem?\newlineChoose 11 answer:\newline(A) ∫34r(t)dt \int_{3}^{4} r(t) d t \newline(B) ∫04r(t)dt \int_{0}^{4} r(t) d t \newline(C) ∫45r(t)dt \int_{4}^{5} r(t) d t \newline(D) ∫44r(t)dt \int_{4}^{4} r(t) d t
Get tutor helpright-arrow

Posted 8 months ago

Question
The function s(t) s(t) gives the number of students enrolled in a school by time t t (in years).\newlineWhat does ∫1518s′(t)dt=20 \int_{15}^{18} s^{\prime}(t) d t=20 mean?\newlineChoose 11 answer:\newline(A) The rate of change of enrollment is 2020 students per year more in year 1818 than it was in year 1515.\newline(B) There were 2020 more students enrolled in year 1818 than in year 1515 .\newline(C) Between years 1515 and 1818, the cumulative number of years of schooling of all of the enrolled students is 2020 years.\newline(D) There are 2020 students enrolled in year 1818.
Get tutor helpright-arrow

Posted 7 months ago

Question
A water tank is filled at a rate of r(t) r(t) liters per minute (where t t is the time in minutes).\newlineWhat does ∫17r′(t)dt \int_{1}^{7} r^{\prime}(t) d t represent?\newlineChoose 11 answer:\newline(A) The rate at which the tank was filled at t=7 t=7 .\newlineB) The average rate of filling between t=1 t=1 and t=7 t=7 .\newline(C) The change in the rate of filling between t=1 t=1 and t=7 t=7 .\newline(D) The amount of water filled between t=1 t=1 and t=7 t=7 .
Get tutor helpright-arrow

Posted 8 months ago

Question
The base of a solid is the region enclosed by the graphs of y=sin⁡(x) y=\sin (x) and y=4−x y=4-\sqrt{x} , between x=2 x=2 and x=7 x=7 .\newlineCross sections of the solid perpendicular to the x x -axis are rectangles whose height is 2x 2 x .\newlineWhich one of the definite integrals gives the volume of the solid?\newlineChoose 11 answer:\newline(A) ∫27[(4−x)2−sin⁡2(x)]dx \int_{2}^{7}\left[(4-\sqrt{x})^{2}-\sin ^{2}(x)\right] d x \newline(B) ∫27[sin⁡(x)+x−4]⋅2xdx \int_{2}^{7}[\sin (x)+\sqrt{x}-4] \cdot 2 x d x \newline(C) ∫27[sin⁡2(x)−(4−x)2]dx \int_{2}^{7}\left[\sin ^{2}(x)-(4-\sqrt{x})^{2}\right] d x \newline(D) ∫27[4−x−sin⁡(x)]⋅2xdx \int_{2}^{7}[4-\sqrt{x}-\sin (x)] \cdot 2 x d x
Get tutor helpright-arrow

Posted 8 months ago

Question
Evaluate the integral.\newline∫−2x3cos⁡(3x)dx \int-2 x^{3} \cos (3 x) d x \newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
Evaluate the integral.\newline∫−4x233xdx \int-4 x^{2} 3^{3 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
Evaluate the integral.\newline∫−4x2e4xdx \int-4 x^{2} e^{4 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
∑n=1∞3nn2n!\sum_{n=1}^{\infty}\frac{3^{n}n^{2}}{n!}
Get tutor helpright-arrow

Posted 8 months ago

Question
f)lim⁡n→+∞[2n+(35)n] f) \lim_{n \to +\infty}\left[\frac{2}{n}+\left(\frac{3}{5}\right)^n\right]
Get tutor helpright-arrow

Posted 8 months ago

Question
∫1cos⁡(x)2 dx\int \frac{1}{\cos(x)^{2}}\,dx
Get tutor helpright-arrow

Posted 8 months ago

View all (198)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant