Evaluate the following integrals. ∫x10xdx

Question

Evaluate the following integrals.  ∫x10xdx

Jillian Edgerton · Accepted Answer

Step 1Given: I=∫x(10)xdxfor evaluating given integral, we will use integral by parts theoremaccording to this theorem∫f′(x)g(x)dx=g(x)∫f′(x)dx−∫[(g′(x))∫f′(x)dx]dx+cStep 2so,I=∫(x)(10x)dx=x∫(10x)dx−∫[1∫10xdx]dx(∵∫axdx=axln⁡a+c)=x(10xln⁡10)−∫(10xln⁡10)dx+c=x(10x)ln⁡10−1ln⁡10(10xln⁡10)+chence, given integral is x(10x)ln⁡10−10x(ln⁡10)2+c.

Andrew Reyes · Accepted Answer

∫x⋅10xdxIntegration piece by piece: ∫fg′=fg−∫f′gf=x,g′=10x=x⋅10xln⁡(10)−∫10xln⁡(10)dx∫10xln⁡(10)dxLet&#039;s apply linearity:=1ln⁡(10)∫10xdx∫10xdxIntegral of exponential function:∫axdx=axln⁡(a) at a=10:=10xln⁡(10)1ln⁡(10)∫10xdx=10xln2(10)x⋅10xln⁡(10)−∫10xln⁡(10)dx=x⋅10xln⁡(10)−10xln2(10)∫x10xdx=x⋅10xln⁡(10)−10xln2(10)+CLet&#039;s rewrite / simplify:=(ln⁡(10)x−1)⋅10xln2(10)+C

Vasquez · Accepted Answer

∫x×10xdx Prepare for integration by parts u=x dv=10xdx du=dx v=10xln⁡(10) x×10xln⁡(10)−∫10xln⁡(10)dx x×10xln⁡(10)−1ln⁡(10)×∫10xdx x×10xln⁡(10)−1ln⁡(10)×10xln⁡(10) Simplify x×10xln⁡(10)−10xln⁡(10)2 Add C Answer: x×10xln⁡(10)−10xln⁡(10)2+C