Use substitution to find the indefinite integral. ∫−4xx2+3dx

Question

Use substitution to find the indefinite integral.  ∫−4xx2+3dx

Jacob Homer · 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  Solve the indefinite integral ∫−4xx2+3dx using substitution method. Substitute x2+3 as t in the given integral.  t=x2+3  dtdx=d(x2+3)dx  dtdx=2x  dx=dt2x  Substitute value of dx=dt2x in integration ∫−4xx2+3dx we get ∫−4xt(dt2x)=∫−2tdt  Step 3  Solve the integral ∫−2tdt for t we get  ∫−2tdt=−2ln|t|+C...(1)  Substitute value of t=x2+3 in equation (1)  −2ln|t|+C=−2ln|x2+3|+C  Therefore, integration of ∫−4xx2+3dx is −2ln|x2+3|+C

trisanualb6 · Accepted Answer

It is required to calculate:  ∫−4xx2+3dx  Substitution u=x2+3⇒dudx=2x⇒dx=12xdu:  =−2∫1udu  Now we calculate:  ∫1udu  This is the well-known tabular integral:  =ln⁡(u)  We substitute the already calculated integrals:  −2∫1udu  =−2ln⁡(u)  Reverse replacement u=x2+3:  =−2ln⁡(x2+3)  Problem solved:  ∫−4xx2+3dx  =−2ln⁡(x2+3)+C

nick1337 · Accepted Answer

∫−4xx2+3dx Transform the expression ∫−2tdt Use properties of integrals −2×∫1tdt Evaluate the integral −2ln⁡(|t|) Substitute back −2ln⁡(|x2+3|) Calculate the absolute value −2ln⁡(x2+3) Add C Solution −2ln⁡(x2+3)+C

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

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