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:

Question\newlineEvaluate ∫x2+1x2−3 dx \int \frac{x^{2}+1}{x^{2}-3} \, dx \newlineProvide your answer below:

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. Question\newlineEvaluate ∫x2+1x2−3 dx \int \frac{x^{2}+1}{x^{2}-3} \, dx \newlineProvide your answer below:
  1. Simplify Integrand: Simplify the integrand if possible.\newlineThe integrand (x2+1)/(x2−3)(x^2 + 1) / (x^2 - 3) cannot be simplified by factoring or partial fractions since the numerator is not of higher degree than the denominator and there are no common factors to cancel out.
  2. Split into Two Fractions: Split the integrand into two separate fractions.\newlineWe can write the integrand as the sum of two separate fractions: x2x2−3+1x2−3\frac{x^2}{x^2 - 3} + \frac{1}{x^2 - 3}.
  3. Simplify First Fraction: Simplify the first fraction.\newlineThe first fraction x2x2−3\frac{x^2}{x^2 - 3} can be simplified to 1+3x2−31 + \frac{3}{x^2 - 3} by long division or by adding and subtracting 33 in the numerator.
  4. Rewrite with Simplified Fractions: Rewrite the integral with the simplified fractions.\newlineNow we can rewrite the integral as the sum of two simpler integrals: ∫1 dx+∫3x2−3 dx\int 1 \, dx + \int \frac{3}{x^2 - 3} \, dx.
  5. Integrate First Part: Integrate the first part ∫1 dx\int 1 \, dx. The integral of 11 with respect to xx is simply xx.
  6. Integrate Second Part: Integrate the second part ∫3x2−3 dx\int \frac{3}{x^2 - 3} \, dx. This integral is of the form ∫duu2−a2\int \frac{du}{u^2 - a^2}, which is a standard integral that can be solved using the substitution method or by recognizing it as a form of the inverse hyperbolic function integral. Let's use substitution. Let u=x2−3u = x^2 - 3, then du=2x dxdu = 2x \, dx. Since we don't have a 2x2x in our integral, we need to adjust for that.
  7. Adjust for Substitution: Adjust the integral to match the substitution.\newlineWe multiply and divide the integrand by 22 to get the correct form for substitution: ∫(32)⋅2u du\int(\frac{3}{2}) \cdot \frac{2}{u} \, du.
  8. Perform Substitution: Perform the substitution.\newlineNow we substitute u=x2−3u = x^2 - 3 into the integral to get ∫(32)⋅2/(u) du=(32)∫duu\int(\frac{3}{2}) \cdot 2 / (u) \, du = (\frac{3}{2}) \int\frac{du}{u}.
  9. Integrate using Natural Logarithm: Integrate using the natural logarithm.\newlineThe integral of 1u\frac{1}{u} with respect to uu is ln⁡∣u∣\ln|u|, so we have (32)ln⁡∣u∣+C(\frac{3}{2}) \ln|u| + C, where CC is the constant of integration.
  10. Substitute Back and Combine: Substitute back for uu. Substituting back for u=x2−3u = x^2 - 3, we get (32)ln⁡∣x2−3∣+C(\frac{3}{2}) \ln|x^2 - 3| + C.
  11. Substitute Back and Combine: Substitute back for uu. Substituting back for u=x2−3u = x^2 - 3, we get (32)ln⁡∣x2−3∣+C(\frac{3}{2}) \ln|x^2 - 3| + C.Combine the results from Step 55 and Step 1010. The final answer is the sum of the integrals from Step 55 and Step 1010, which is x+(32)ln⁡∣x2−3∣+Cx + (\frac{3}{2}) \ln|x^2 - 3| + C.

More problems from Evaluate definite integrals using the power 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 (52)

Related topics

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