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:

F(x)=int_(1)^(2x)(t^(2))/(t^(2)+1)dt F^(')(x)=

F(x)=∫12xt2t2+1dt F(x)=\int_{1}^{2 x} \frac{t^{2}}{t^{2}+1} d t \newlineF′(x)= F^{\prime}(x)=

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

Full solution

Q. F(x)=∫12xt2t2+1dt F(x)=\int_{1}^{2 x} \frac{t^{2}}{t^{2}+1} d t \newlineF′(x)= F^{\prime}(x)=
  1. Apply Fundamental Theorem of Calculus: Use the Fundamental Theorem of Calculus Part 11 which states that if F(x)=∫axf(t) dtF(x) = \int_{a}^{x}f(t)\,dt, then F′(x)=f(x)F'(x) = f(x). However, since the upper limit of the integral is 2x2x, we need to apply the chain rule.
  2. Differentiate Upper Limit: Differentiate the upper limit of the integral with respect to xx, which is 2x2x. The derivative is 22.
  3. Apply Chain Rule: Now apply the chain rule: F′(x)=f(2x)⋅d(2x)dxF'(x) = f(2x) \cdot \frac{d(2x)}{dx}. Substitute t=2xt = 2x into the integrand and multiply by the derivative of 2x2x.
  4. Substitute and Multiply: F′(x)=(2x)22x2+1×2F'(x) = \frac{(2x)^2}{2x^2+1} \times 2.
  5. Simplify Expression: Simplify the expression: F′(x)=4x24x2+1×2F'(x) = \frac{4x^2}{4x^2+1} \times 2.
  6. Final Result: F′(x)=8x24x2+1F'(x) = \frac{8x^2}{4x^2+1}.

More problems from Evaluate definite integrals using the power rule

Question
The number of subscribers to a magazine is changing at a rate of r(t) r(t) subscribers per month (where t t is time in months).\newlineWhat does ∫810r′(t)dt=7 \int_{8}^{10} r^{\prime}(t) d t=7 mean?\newlineChoose 11 answer:\newline(A) The rate of change of number of subscribers increased by 77 subscribers per month between t=8 t=8 and t=10 t=10 months.\newline(B) As of month 1010, the magazine had 77 subscribers.\newline(C) The number of subscribers increased by 77 between t=8 t=8 and t=10 t=10 months.\newline(D) The average rate of change in subscribers between month 88 and month 1010 was 77 subscribers per month.
Get tutor helpright-arrow

Posted 9 months ago

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) ∫04r(t)dt \int_{0}^{4} r(t) d t \newline(B) ∫45r(t)dt \int_{4}^{5} r(t) d t \newline(C) ∫34r(t)dt \int_{3}^{4} r(t) d t \newline(D) ∫44r(t)dt \int_{4}^{4} r(t) d t
Get tutor helpright-arrow

Posted 9 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[sin⁡(x)+x−4]⋅2xdx \int_{2}^{7}[\sin (x)+\sqrt{x}-4] \cdot 2 x d x \newline(B) ∫27[sin⁡2(x)−(4−x)2]dx \int_{2}^{7}\left[\sin ^{2}(x)-(4-\sqrt{x})^{2}\right] d x \newline(C) ∫27[4−x−sin⁡(x)]⋅2xdx \int_{2}^{7}[4-\sqrt{x}-\sin (x)] \cdot 2 x d x \newline(D) ∫27[(4−x)2−sin⁡2(x)]dx \int_{2}^{7}\left[(4-\sqrt{x})^{2}-\sin ^{2}(x)\right] d x
Get tutor helpright-arrow

Posted 9 months ago

Question
(1×((12)n−1))/((12)−1)\left(1\times\left(\left(\frac{1}{2}\right)^n-1\right)\right)\bigg/\left(\left(\frac{1}{2}\right)-1\right)
Get tutor helpright-arrow

Posted 8 months ago

Question
Solve∫1(x−2)(x2+x+1) dx\int \frac{1}{(x-2)(x^{2}+x+1)}\,dx.
Get tutor helpright-arrow

Posted 8 months ago

Question
What is the integral of the function f(x)=sin⁡(2x) f(x) = \sin(2x) ?\newline−12cos⁡(x)+C \frac{-1}{2}\cos(x) + C \newline12sin⁡(x)+C \frac{1}{2}\sin(x) + C \newline−12cos⁡(2x)+C \frac{-1}{2}\cos(2x) + C \newline12sin⁡(2x)+C \frac{1}{2}\sin(2x) + C
Get tutor helpright-arrow

Posted 8 months ago

Question
What is the integral of the function  f(x)=sin(2x)\ f(x) = sin(2x)?\newline−12cos⁡(x)+C \frac{-1}{2}\cos(x) + C \newline12sin⁡(x)+C \frac{1}{2} \sin(x) + C \newline−12cos⁡(2x)+C \frac{-1}{2}\cos(2x) + C \newline12sin⁡(2x)+c \frac{1}{2} \sin(2x) + c
Get tutor helpright-arrow

Posted 8 months ago

Question
Suppose ∫24f(2x) dx=10 \int_{2}^{4} f(2x) \, dx = 10 . Then ∫48f(u) du= \int_{4}^{8} f(u) \, du =
Get tutor helpright-arrow

Posted 8 months ago

Question
Find ddt∫2t4ex2dx\frac{d}{dt}\int_{2}^{t^{4}}e^{x^{2}}dx
Get tutor helpright-arrow

Posted 8 months ago

Question
Evaluate and write the answer in scientific notation: \newline(1.48×103)(7×104)÷8.2×1012(1.48\times10^{3})(7\times10^{4})\div8.2\times10^{12}
Get tutor helpright-arrow

Posted 8 months ago

View all (25)

Related topics

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