Question

Evaluate the indefinite integral in each item

1. ∫e^2x+1dx
2. ∫1/(1+e^x) dx
3. ∫e^2x/(e^x+3) dx
4. ∫4(10^{2x^2}) dx
5. ∫x²/(x³+5) dx

Answer 1

1. $\int e^{2x+1}dx=\frac12\int e^{2x+1}d(2x+1)=\frac12e^{2x+1}+C,C\in{\mathbb{R}}$ .
2. $\int\frac{dx}{e^x+1}$. We make the change: $z=e^x$. We point out that $z>0$. We receive: $\int\frac{dz}{z(z+1)}=\int(\frac{1}{z}-\frac{1}{z+1})dz=ln\,z-ln(z+1)+C=x-ln(e^x+1)+C,C\in{\mathbb{R}}$
3. $\int\frac{e^{2x}}{e^x+1}dx$ . We make the change $z=e^x$ and receive: $\int\frac{zdz}{z+1}=\int(1-\frac{1}{z+1})dz=z-ln(z+1)+C=e^x-ln(e^x+1)+C,C\in{\mathbb{R}}$
4. $\int 4(10^{2x^2})dx=\int4e^{2x^2ln2}dx.$ The good expression for this integral in terms of elementary functions is not known.
5. $\int\frac{x^2}{x^3+5}dx=\int\frac{1}{3(x^3+5)}d(x^3+5)=3ln(x^3+5)+C,C\in{\mathbb{R}}.$