The integral ∫$$01/a$$ log⁡ (1+ax)1+a2x2dx(a$$$$0) is equal to

Question

The integral ∫$$01/a$$ log⁡ (1+ax)1+a2x2dx(a$$>$$0) is equal to

Infinity Learn · Accepted Answer

Put $$ax=t$$, the given integral reduces $$to1a$$∫$$01$$ log⁡ (1+t)1+t2dt=1a$$∫$$0$$π$$/4$$ log⁡ $$(1+$$$$tan$$⁡ $$u)1+tan2$$⁡ $$usec2$$⁡ $$udu(t$$−$$tan$$⁡ $$u)=1a$$∫$$0$$π$$/4$$ log⁡ $$(1+$$$$tan$$⁡ $$u)du=1a$$∫$$0$$π$$/4$$ log⁡$$1+$$$$tan$$⁡π$$4$$−$$udu=1a$$∫$$0$$π$$/4$$ log⁡ $$1+1$$−$$tan$$⁡ $$u1+$$$$tan$$⁡ $$udu=1a$$∫$$0$$π$$/4$$ [log⁡ $$2$$−log⁡ $$(1+$$$$tan$$⁡ $$u)]du⇒$$2a$$∫$$0$$π$$/4$$ log⁡ (1+$$$$tan$$⁡ $$u)du=1a(log$$⁡ $$2)π$$4$$⇒$$1a$$∫$$0$$π$$/4$$ log⁡(1+$$$$tan$$⁡ $$u)du=1a(log$$⁡ $$2)π$$8$$

The integral $\int_{0}^{1 / a} \frac{\log (1 + a x)}{1 + a^{2} x^{2}} d x (a > 0)$ is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

The integral ∫01/a log⁡ (1+ax)1+a2x2dx(a>0) is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

The integral $\int_{0}^{1 / a} \frac{\log (1 + a x)}{1 + a^{2} x^{2}} d x (a > 0)$ is equal to