Definite integrals by substitution practice problems
Definite integrals by substitution, changing the limits along with the variable. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \int_{0}^{1} \frac{3 x}{x^{2} + 4}\, dx \).
Evaluate \( \displaystyle \int_{0}^{3} \frac{2 x}{x^{2} + 3}\, dx \).
Evaluate \( \displaystyle \int_{0}^{\frac{\pi}{2}} \frac{4 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}\, dx \).
Evaluate \( \displaystyle \int_{0}^{\frac{\pi}{2}} \frac{5 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}\, dx \).
Evaluate \( \displaystyle \int_{0}^{1} 5 x e^{x^{2}}\, dx \).
Evaluate \( \displaystyle \int_{0}^{2} \frac{5 x^{2}}{\sqrt{x^{3} + 4}}\, dx \).
Evaluate \( \displaystyle \int_{1}^{e} \frac{3 \ln{\left(x \right)}^{4}}{x}\, dx \).
Evaluate \( \displaystyle \int_{1}^{e^{2}} \frac{3 \ln{\left(x \right)}^{4}}{x}\, dx \).
Evaluate \( \displaystyle \int_{0}^{\frac{\pi}{2}} 3 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx \).
Evaluate \( \displaystyle \int_{1}^{e} \frac{4 \ln{\left(x \right)}^{2}}{x}\, dx \).