Integration by parts

28 practice questionsIntegral CalculusStep-by-step solutions

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Given $I = \int_1^{9} \sqrt{x}\ln x\,dx$ . What is the final simplified expression of $\int \sqrt{x}\ln x\,dx$ , before adding the limits?

\frac{2}{3}x^{\frac32}(\ln x - \frac{2}{3})

x^{\frac12}(\frac32\ln x - \frac94)

x^{\frac32}(\ln x - 1)

\frac32 x^{-\frac12}(\ln x - \frac23)

E.None

Determine $\int \ln(x^2+1)\,dx$ .

x\ln(x^2+1)-2x+2\tan^{-1}x+C

x\ln(x^2+1)+2x-2\tan^{-1}x+C

x\ln(x^2+1)-2x-2\tan^{-1}x+C

2x\ln(x^2+1)-2x+2\tan^{-1}x+C

Determine $\int x^3 e^x\,dx$ (repeated by-parts / reduction pattern).

(x^3-3x^2+6x-6)e^x+C

(x^3+3x^2+6x+6)e^x+C

(x^3-3x^2+6x-6)e^{-x}+C

\dfrac{x^4}{4}e^x+C

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