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I am trying to do this integration but somehow I don't get the right answer: $$\int(4(2x+1)^7)dx$$ $$u=2x+1 \space \therefore \dfrac{du}{dx}=2 \space \therefore dx=\dfrac{1}{2}du $$ $$4\times 2\int u^7\times \dfrac{1}{2}du$$ $$4\int u^7du = 4\dfrac{u^8}{8}=\dfrac{1}{2}(2x+1)^8+C$$ However it should be $\dfrac{1}{4}(2x+1)^8 + C$ |
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This is intended to clarify my first two comments. Let's omit the factor of $4$, which is unrelated to your question. Then you want to evaluate $$\int(2x+1)^7\;dx$$ Here's one way to view substitution. We know that $$\int(boop)^7\;d(boop) \; = \; \left(\frac{1}{8}\right)(boop)^8 + C$$ A version of this that is close to what you have is $$\int(2x+1)^7\;d(2x+1)$$ However, you have $dx$ and the integral I just wrote has $d(2x+1)$. But since $d(2x+1) = 2\,dx,$ we can get $d(2x+1)$ to show up this way: $$\int(2x+1)^7\;dx \; = \; \int(2x+1)^7\;\left(\frac{1}{2}\right)\cdot 2\,dx$$ $$ = \; \int(2x+1)^7\;\left(\frac{1}{2}\right)\cdot d(2x+1) \; = \; \frac{1}{2}\int(2x+1)^7 \; d(2x+1)$$ Now here's how your $u$-substitution method works in light of what I did above. You want to get $$\int u^7 \,du$$ to show up. Obviously, $u = 2x+1$, so to get $du$ to appear we need to see what $du$ is. As you've shown, $du = 2\,dx.$ At this point you can either multiply and divide by $2$ to get $2\,dx$ to show up (the approach I took above), or you can simply substitute $\frac{1}{2}du$ in place of $dx$ (what you did). But if you choose the direct substitution method, you don't have to multiply by $2$ (where you made your error), since you're just replacing equals with equals. |
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Let me solve this question step-by-step for you. $$\int4(2x+1)^7dx$$ $$u=2x+1$$ $$\frac{du}{dx}=2 \therefore dx=\frac12du$$ $$\therefore \int4(2x+1)^7dx=4\int u^7 * \frac12du=2\int u^7 du$$ This becomes: $$2(\frac{u^8}8) + c=\frac{u^8}4+c$$ $$=\frac{(2x+1)^8}4+c$$ |
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In general, for a derivable function $\,f(x)\,$ , we have $$\int f'(x)f(x)^ndx=\frac{f(x)^{n+1}}{n+1}+C$$ In our case, $\,f(x)=(2x+1)\,\,,\,\,f'(x)=2\,$ , so: $$\int 4(2x+1)^7\,dx=2\int(2\,dx)(2x+1)^7=2\frac{(2x+1)^8}{8}+C=\frac14(2x+1)^8+C$$ |
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