Substitution with a given u practice problems
Indefinite integrals with a given substitution u: change variables, integrate, substitute back. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \int \frac{\ln{\left(x \right)}^{3}}{x}\, dx \) using the substitution \( \displaystyle u = \ln{\left(x \right)} \).
Evaluate \( \displaystyle \int 2 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \) using the substitution \( \displaystyle u = \sin{\left(2 x \right)} \).
Evaluate \( \displaystyle \int 2 x \left(x^{2} + 2\right)^{2}\, dx \) using the substitution \( \displaystyle u = x^{2} + 2 \).
Evaluate \( \displaystyle \int \frac{e^{3 x}}{e^{3 x} + 2}\, dx \) using the substitution \( \displaystyle u = e^{3 x} + 2 \).
Evaluate \( \displaystyle \int 4 \left(e^{2 x} + 3\right)^{6} e^{2 x}\, dx \) using the substitution \( \displaystyle u = e^{2 x} + 3 \).
Evaluate \( \displaystyle \int 3 \sin^{6}{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \) using the substitution \( \displaystyle u = \sin{\left(2 x \right)} \).
Evaluate \( \displaystyle \int \frac{3 \ln{\left(x \right)}^{4}}{x}\, dx \) using the substitution \( \displaystyle u = \ln{\left(x \right)} \).
Evaluate \( \displaystyle \int \frac{6 e^{2 x}}{e^{2 x} + 2}\, dx \) using the substitution \( \displaystyle u = e^{2 x} + 2 \).
Evaluate \( \displaystyle \int 3 \sin^{3}{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \) using the substitution \( \displaystyle u = \sin{\left(2 x \right)} \).
Evaluate \( \displaystyle \int 6 \sin^{4}{\left(4 x \right)} \cos{\left(4 x \right)}\, dx \) using the substitution \( \displaystyle u = \sin{\left(4 x \right)} \).