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Ten direct integrals and reverse chain rules

Find the following integrals:

(a) ∫x(x+3) dx\displaystyle\int \sqrt{x}(x+3)\,dx

(b) ∫(y+1y)2 dy\displaystyle\int \left(\sqrt y+\dfrac1{\sqrt y}\right)^2\,dy

(c) ∫x+1(x2+2x+3)3 dx\displaystyle\int \dfrac{x+1}{(x^2+2x+3)^3}\,dx

(d) ∫ln⁡xx dx\displaystyle\int \dfrac{\ln x}x\,dx

(e) ∫3t1−t dt\displaystyle\int \dfrac{3t}{1-t}\,dt

(f) ∫1xln⁡x dx\displaystyle\int \dfrac1{x\ln x}\,dx

(g) ∫ex1+ex dx\displaystyle\int \dfrac{e^x}{1+e^x}\,dx

(h) ∫exx dx\displaystyle\int \dfrac{e^{\sqrt x}}{\sqrt x}\,dx

(i) ∫e1/xx2 dx\displaystyle\int \dfrac{e^{1/x}}{x^2}\,dx

(j) ∫20172017x dx\displaystyle\int 2017^{2017x}\,dx.

CJC · Discussion · Question 3

Look for a derivative of the inner expression; expand only when it makes the integrand simpler.

Read the worked solution