Use integration by parts to evaluate the integral. (Use C for the constant of integration. ∫ln⁡(5x)3x2dx

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Use integration by parts to evaluate the integral. (Use C for the constant of integration.  ∫ln⁡(5x)3x2dx

hotcoal3z · Accepted Answer

Step 1  Given , the integral  ∫ln⁡(5x)3x2dx  We have to evaluate the integral by using by parts formula.  Step 2  By-parts formula  ∫f(x)g(x)dx=f(x)∫g(x)dx−∫{ddxf(x)∫g(x)dx}dx+c  Where c is constant of integration.  Step 3  Hence,  ∫ln⁡(5x)3x2dx=ln⁡(5x)∫13x2dx−∫{ddx{ln⁡(5x)}∫13x2dx}dx+c  =ln⁡(5x)(−13x)−∫{55x(−13x)}dx+c  =−13xln⁡(5x)−∫(−13)dx+c  =−13xln⁡(5x)+13x+c  =13x{1−ln⁡(5x)}+c

giskacu · Accepted Answer

Step 1: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx.  13∫ln⁡5xx2dx  Step 2: Use Integration by Parts on ∫ln⁡5xx2dx.  Let u=ln⁡5x,dv=1x2,du=1xdx,v=−1x  Step 3: Substitute the above into uv−∫vdu.  13(−ln⁡5xx−∫−1x2dx)  Step 4: Use Power Rule: ∫xndx=xn+1n+1+C.  −ln⁡5x3x−13x  Step 5: Add constant.  −ln⁡5x3x−13x+C

Use integration by parts to evaluate the integral. (Use C

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