Porisma

Worked solution

Reverse a differentiated composite

Differentiate (5x2+2x+1)8(5x^2+2x+1)^8 with respect to xx. Hence find ∫(5x+1)(5x2+2x+1)7 dx\int(5x+1)(5x^2+2x+1)^7\,dx.

CJC · Lecture · Example 1

Differentiate

ddx(5x2+2x+1)8=8(5x2+2x+1)7(10x+2)=16(5x+1)(5x2+2x+1)7\frac{d}{dx}(5x^2+2x+1)^8=8(5x^2+2x+1)^7(10x+2)=16(5x+1)(5x^2+2x+1)^7.

How did this part go?

Hence integrate

∫(5x+1)(5x2+2x+1)7 dx=116∫16(5x+1)(5x2+2x+1)7 dx=116(5x2+2x+1)8+C\int(5x+1)(5x^2+2x+1)^7\,dx=\frac1{16}\int16(5x+1)(5x^2+2x+1)^7\,dx=\frac1{16}(5x^2+2x+1)^8+C.

How did this part go?

Common mistakes

Reverse the complete chain rule, including the factor 16, and include an arbitrary constant.

Read the hint